Analysis
Andrew Monnot
Contents
1 Measure Spaces
1.1 Measure Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 More on Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
2
3
6
2 Probability Spaces
8
3 Analysis on Other Structures
3.1 Topological and Uniform Spaces
3.2 Metric Spaces . . . . . . . . . .
3.3 Normed Spaces . . . . . . . . .
3.4 The Bochner Integral . . . . . .
3.5 Lp Spaces . . . . . . . . . . . .
3.6 Inner Product Spaces . . . . . .
3.7 Riesz Spaces . . . . . . . . . . .
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12
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16
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19
20
Algebras
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26
26
5 Analysis on Manifolds
5.1 Differentiable Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Riemannian Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
28
32
References
34
4 Dual Spaces and Operator
4.1 Dual Spaces . . . . . . .
4.2 Operator Algebras . . .
4.3 Spectral Theory . . . . .
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1
Measure Spaces
1.1
Measure Spaces
Definition 1.1. A σ-algebra is a nonempty collection Σ of sets such that
(i) An ∈ Σ for n ∈ N ⇒ ∪n∈N An ∈ Σ
(ii) A ∈ Σ ⇒ AC ∈ Σ.
It follows that a σ-algebra is closed under countable intersections. Also if Σ is a
σ-algebra over a nonempty set X (containing subsets of X), we have that A, AC ∈ Σ
and hence A ∩ AC = ∅ ∈ Σ and thus ∅C = X ∈ Σ. Hence a topology on X induces a
σ-algebra on X by closing the topology under complementation. Such a σ-algebra Σ(τ )
is called the Borel σ-algebra on X, and its sets are called Borel sets.
Definition 1.2. A measure on a σ-algebra on X, Σ, is a map µ : Σ → R∗ (where
R∗ = [0, ∞]) satisfying
(i) µ(∅) = 0
(ii) µ(∪n∈N An ) =
P
n∈N
µ(An ) for disjoint sets {An } ⊆ Σ.
Definition 1.3. A measure space is a collection (X, Σ, µ) where Σ is a σ-algebra on
X and µ is a measure on Σ. We call a pair (X, Σ) a measurable space.
A measure µ is said to be finite if µ(X) < ∞ (and is thus finite on all A ∈ Σ since
µ(X) = µ(A) + µ(AC )). µ is σ-finite if X = ∪n∈N Bn where µ(Bn ) < ∞ for all n.
Example 1.4. P
Let X be nonempty and Σ = 2X . If f : X → [0, ∞], then we have a
measure µ(A) = x∈A f (x). When f (x) = 1 for all X the measure is called the counting
measure. The Dirac measure is generated from the function fx0 defined by fx0 (x0 ) = 1
and fx0 (x) = 0 for x 6= x0 .
Proposition 1.5. Let (X, Σ, µ) be a measure space.
(a) (Monotonicity) If A, B ∈ Σ and A ⊆ B, then µ(A) ≤ µ(B).
P
(b) (σ-Subadditivity) If {An } ⊆ Σ, then µ(∪n∈N An ) ≤ n∈N µ(An ).
(c) (Continuity from below)
µ(∪n∈N An ) = limn→∞ µ(An ).
If {An } ⊆ Σ such that A1 ⊆ A2 ⊆ · · · , then
(d) (Continuity from above) If {An } ⊆ Σ such that A1 ⊇ A2 ⊇ · · · and µ(A1 ) <
∞, then µ(∩n∈N An ) = limn→∞ µ(An ).
Proof. (a) We have
µ(B) = µ(B) + µ(A) − µ(A) = µ(A) + µ(B − A) ≥ µ(A).
n
(b) Let B1 = A1 and Bn = An − ∪n−1
i=1 Ai . Then {Bn } are disjoint sets and ∪1 Bi =
n
∪1 Ai . Thus
!
!
[
[
X
X
µ
An = µ
Bn =
µ(Bn ) ≤
µ(An ).
n∈N
n∈N
n∈N
2
n∈N
(c) With A0 = ∅, we have
!
k
[
X
X
µ
An =
µ(An − An−1 ) = lim
µ(An − An−1 ) = lim µ(Ak ).
n∈N
k→∞
n∈N
n=1
k→∞
(d) Here we let Bn = A1 − An . It follows that B1 ⊆ B2 ⊆ · · · and that µ(A1 ) =
µ(Bn ) + µ(An ). We also have that ∪n∈N Bn = A1 − ∩n∈N An . Thus using (c) we obtain
!
!
[
\
lim (µ(A1 ) − µ(An )) = lim µ(Bn ) = µ
Bn = µ(A1 ) − µ
An
n→∞
n→∞
n∈N
n∈N
from which the result follows since µ(A1 ) < ∞.
Definition 1.6. A function µ∗ : 2X → R∗ is an outer measure if it satisfies condition
(i) of a measure, monotonicity, and subadditivity (hence measures are outer measures).
A set A ∈ 2X is µ∗ -measurable for an outer measure µ∗ if
µ∗ (X) = µ∗ (X ∩ A) + µ∗ (X ∩ AC )
for all X ∈ 2X .
A set A ∈ Σ is called a null set with respect to µ if µ(A) = 0. A set A ∈ Σ is an
atom with respect to µ if µ(A) > 0 and µ(B) = 0 for any subset B ⊆ A such that
µ(B) < µ(A). A measure whose domain includes all subsets of null sets is said to be
complete.
Theorem 1.7. (Carathéodory) If µ∗ is an outer measure on X and M is the collection
of µ∗ -measurable sets, then M is a σ-algebra and the restriction of µ∗ to M is a complete
measure.
Definition 1.8. Let (X, Σ1 ) and (Y, Σ2 ) be measurable spaces. A function f : X → Y
is measurable if when A ∈ Σ2 we have that f −1 (A) ∈ Σ1 .
Hence if f : X → Y is a continuous map between topological spaces, then it is also a
measurable map between the induced measurable spaces.
1.2
Integration
From now on we assume to be working in a measure space (X, Σ, µ).
Definition 1.9. A simple (or piecewise constant) function on X is a function of
the form
n
X
φ(x) =
ci χAi (x)
i=1
where ci ∈ R and
χAi (x) =
1 if x ∈ Ai
0 if x ∈
/ Ai
is the indicator function on some Ai ∈ Σ such that {Ai } is a partition of X.
3
Theorem 1.10. Let f : (X, Σ) → (R, BR ) be a nonnegative measurable function where
BR is the Borel σ-algebra on R. Then there is a sequence {φn } of simple functions such
that 0 ≤ φ1 ≤ · · · ≤ f and such that φn → f pointwise and φn → f uniformly on any set
on which f is bounded.
Proof. For φn we will set φn (x) = 2n whenever f (x) ≥ 2n . When f (x) ≤ 2n we
subdivide vertically into k half open intervals and assign φn (x) = k2−n when k2−n ≤
f (x) ≤ (k + 1)2−n . The last interval will be (2n − 2−n , 2n ], which means k goes up to
22n − 1. Hence we define
2n −1
2X
φn =
k2−n χEnk + 2n χFn
k=0
where Enk = f −1 ((k2−n , (k + 1)2−n ]) and Fn = f −1 ((2n , ∞]). It easily follows that
φn ≤ φn+1 and that 0 ≤ f − φn ≤ 2−n → 0 when f ≤ 2n .
Let L+ denote the space of all measurable functions from (X, Σ, µ) to [0, ∞], B[0,∞] .
P
Definition 1.11. Let φ = ni=1 ci χAi be a simple function in L+ . We define the simple
integral of φ as
Z
n
X
φ dµ =
ci µ(Ai ).
X
i=1
If f : X → [0, ∞] is any function in L+ , then we define the general integral of f as
Z
Z
f dµ = sup
φ dµ : 0 ≤ φ ≤ f
φ∈L+
X
which exists by the previous theorem. If f takes positive and negative values, let us define
f + (x) = max{0, f (x)}
f − (x) = min{0, −f (x)}.
Both of these are nonnegative functions and f = f + −f − . If each is in L+ and at least
one is not infinite when integrated, then we say f is integrable and define the general
integral of f as
Z
Z
Z
+
f dµ =
f dµ −
f − dµ.
X
R
X
X
R
We will often write X f dµ = f . It turns out that f is integrable iff |f | is integrable.
This is easy to see since |f | = f + + f − . We will denote the class of integrable functions
over (X, Σ, µ) as L1 (X, µ). When X = R and µ is the Lebesgue measure defined by
λ([a, b]) = b − a, we denote L1 (X, µ) simply by L1 , the class of Lebesgue integrable
functions.
Theorem 1.12. (Monotone Convergence Theorem) If {fn } is a sequence in L+
such that fn ≤ fn+1 for all n, and limn→∞ fn = f , then
Z
Z
f = lim
fn .
n→∞
4
R
R
Proof. Since
{f
}
is
monotonically
increasing,
we
have
that
f
≤
f for all n and
n
n
R
R
hence limn→∞ fn ≤ f . Now let α ∈ (0, 1) and En = {x ∈ X : fn (x) ≥ αφ(x)} where
φ is any simple function such that 0 ≤ φ(x) ≤ f . So En is increasing in the sense of
containment, and we have ∪n En = X. Thus
Z
Z
Z
fn ≥
fn ≥ α
φ.
X
R
En
En
R
R
R
So we have limn→∞ En φ = X φ and hence limn→∞ X fn ≥ α X φ. Letting α → 1 and
R
R
taking the supremum over all simple φ, we obtain limn→∞ fn ≥ f .
Lemma 1.13. (Fatou’s Lemma) If {fn } is any sequence in L+ , then
Z
Z
lim inf fn ≤ lim inf fn .
Proof. For each k note that we have inf n≥k fn ≤ fj for j ≥ k. Thus
for j ≥ k, and hence it holds for the infimum:
Z
Z
inf fn ≤ inf fj .
R
inf n≥k fn ≤
R
fj
j≥k
n≥k
Note that inf n≥k fn ≤ inf n≥k+1 fn , so by the monotone convergence theorem we have
Z
Z
Z
lim inf fn = lim
inf fn ≤ lim inf fn .
k→∞
n≥k
We will say that a sentence S((X, Σ, µ)) is true almost everywhere (a.e.) if it is
true except on a set F ∈ Σ for which µ(F ) = 0.
Theorem 1.14. (Dominated Convergence Theorem) Let {fn } be a sequence in
L1 (X, µ) such that fn → f a.e. and g ∈ L1 (X, µ) such that |fn | ≤ g a.e. for all n. Then
f ∈ L1 (X, µ) and
Z
Z
f = lim
n→∞
fn .
Proof. f is measurable since fn → f a.e.. So for an open E ⊆ R one can find a
sequence En ⊆ fn−1 (E) such that f −1 (E) = ∩n En ∈ Σ. We also have |f | ≤ g a.e., so
f ∈ L1 (X, µ).
We must have g is nonnegative by |fn | ≤ g a.e., so g + fn ≥ 0 a.e. and g − fn ≥ 0 a.e.
We can write lim inf fn = f , so by Fatou’s Lemma we have
Z
Z
Z
Z
Z
g + f ≤ lim inf (g + fn ) = g + lim inf fn ,
Z
Z
Z
Z
Z
g − f ≤ lim inf (g − fn ) = g − lim sup fn .
Thus lim inf
R
fn ≥
R
R
f ≥ lim sup fn , and we obtain the result.
5
Definition 1.15. Let (X, M, µ) and (Y, N , ν) be measure spaces. We define the product measure space as the measure space (X × Y, M × N , µ × ν) where M × N , the
product σ-algebra, is the σ-algebra generated by pairs A × B with A ∈ M and B ∈ N
and where µ × ν is the product measure defined additively by
(µ × ν)(A × B) = µ(A)ν(B).
Theorem 1.16. (Fubini-Tonelli) Suppose (X, M, µ) and (Y, N , ν) are σ-finite measure spaces.
R
(a) R(Tonelli) If f ∈ L+ (X × Y ), then the functions g(x) = f (x, y) dν and h(y) =
f (x, y) dµ are in L+ (X) and L+ (Y ) respectively and
Z
Z Z
f d(µ × ν) =
f (x, y) dν dµ
Z Z
=
f (x, y) dµ dν.
(b) (Fubini) If f ∈ L1 (µ × ν), then f ∈ L1 (ν) for a.e. x ∈ X and f ∈ L1 (µ) for a.e.
y ∈ Y . We also have g(x) ∈ L1 (µ) and h(y) ∈ L1 (ν), and the above equality
holds.
1.3
More on Measures
Recall condition (ii) of a measure. This is called σ-additivity after excluding the disjointness requirement. Additivity is the weaker case that holds for finite sequences {An } ⊆ Σ.
A semiring of sets is a collection of sets containing ∅, closed under intersections, and
closed under differences that can be written as finite disjoint unions of other sets in the
semiring.
Definition 1.17. A signed charge on a semiring of sets S is an additive map µ : S →
[−∞, ∞] such that not both ±∞ are in the image and for which µ(∅) = 0. A charge is
a nonnegative signed charge. A signed measure on a semiring of sets S is a σ-additive
signed charge. A measure is hence a signed measure which is also a charge. An outer
measure is nonnegative and satisfies monotonicity, σ-subadditivity, and µ(∅) = 0.
Hence we have
nnn
nnn
n
n
nn
rz nn
charge dl Q
outer measure
KS
QQQQ
QQQQ
QQQQ
Q
measure
mmm
m
m
mmm
mmm
m
m
rz
signed charge
dl P
PPPP
PPPP
PPP
P
signed measure
For topological spaces we can further classify measures on the corresponding Borel
sets.
Definition 1.18. A measure µ on (X, Bτ ) is
6
(i) outer regular if
µ(A) = inf{µ(V ) : V open and A ⊆ V };
(ii) inner regular if
µ(A) = sup{µ(U ) : U closed and U ⊆ A};
(iii) normal if it is outer and inner regular;
(iv) tight if
µ(A) = sup{µ(K) : K is compact and K ⊆ A};
(v) regular if it is finite on compact sets, outer regular, and tight;
(vi) locally finite if every point has a neighborhood of finite measure;
(vii) Radon if it is inner regular and locally finite
for all A ∈ Bτ .
Hence we have
regular
tight
oo
ooo
o
o
oo
s{ oo
outer regular
bj L
oo
ooo
o
o
ooo
s{ oo
measure
KS ck OOO
OOOO
OOOO
O
locally finite
ck O
LLLLL
LLLLLL
LLLLLL
LL
normal
rrr
rrrrr
r
r
r
rrr
t| rrrr
inner regular
KS
OOOO
OOOO
OOO
Radon
In Hausdorff spaces compact implies closed, in which case tight implies inner regular.
In particlar tight and regular are equivalent.
Definition 1.19. Let {fn } be a sequence of real-valued functions on (X, Σ, µ). This
sequence is Cauchy in measure if for every ε > 0 we have that
µ ({x : |fn (x) − fm (x)| ≥ ε}) → 0
as m, n → ∞. The sequence converges in measure to a function f (x) if for every ε > 0
we have that
µ ({x : |fn (x) − f (x)| ≥ ε}) → 0
as m, n → ∞.
One can show that if a sequence of functions is Cauchy in measure, then there is a
measurable function f such that {fn } converges in measure to f . It’s trivial to verify
that convergence a.e. implies convergence in measure.
7
Definition 1.20. Let µ and ν be signed measures. We say µ and ν are mutually
singular, denoted µ ⊥ ν, if there are measurable sets A, B such that X = A t B and for
which µ(B) = 0 and ν(A) = 0. We say µ is absolutely continous with respect to ν,
denoted µ ν, if ν(A) = 0 ⇒ µ(A) = 0.
Theorem 1.21. (Decomposition of Measures)(Hahn-Jordan) If µ is a signed
measure on (X, Σ), then there are measurable sets P, N such that X = P t N and
for which µ(P ) ≥ 0 and µ(N ) ≤ 0. If P 0 , N 0 is another such pair, then µ(P ∆P 0 ) =
µ(N ∆N 0 ) = 0. Furthermore we can write µ = µ+ − µ− for two mutually singular
measures µ+ , µ− .
Correspondingly we can define integration with respect to signed measures:
Z
Z
Z
+
f dµ = f dµ − f dµ− .
2
Probability Spaces
A probability space is a measure space (X, Σ, P ) for which P (X) = 1. Hence finite measure spaces are equivalent to probability spaces since for a finite measure space
(X, Σ, µ), we can define
µ(A)
.
P (A) =
µ(X)
Definition 2.1. Let A and B be measurable sets. We define the conditional probability of A given B as
P (A ∩ B)
P (A|B) =
.
P (B)
A and B are said to be independent if P (A ∩ B) = P (A)P (B).
Hence it follows that if A and B are independent, we have that P (A|B) = P (A).
Definition 2.2. A random variable is a measurable function defined on a probability
space. That is, it is a function f : (X, M, P ) → (Y, N )
A random variable hence induces a measure on its codomain, called the probability
distribution of f , defined by
Pf (A) = P (f −1 (A))
for A ∈ N .
Definition 2.3. Let f be a real-valued random variable on (X, Σ, P ). We define the
expectation of f as
Z
E(f ) =
f dP.
X
Proposition 2.4. Consider two random variables f, g:
(X, M, P )
f
We have that
/
g
(Y, N , Pf )
Z
E(g) =
/
(Rn , BRn , λ) .
Z
g ◦ f dP.
g dPf =
Y
X
8
Proof. If g = χA for some A ∈ N , then we have χA ◦ f = χf −1 (A) and the statement
holds. The general result follows as a limit of simple functions.
Definition 2.5. Let f and g be real-valued random variables. We define the standard
deviation of f as
q
σ(f ) = inf E((f − a)2 ).
a∈R
2
σ (f ) is called the variance of f . We define the covariance of f and g as
Cov(f, g) = E(f g) − E(f )E(g).
Definition 2.6. Let f and g be real-valued random variables with A ∈ BRn . We define
the conditional expectation of f with respect to (g, A) as
Z
Z
1
1
E(f |(g, A)) =
f dP =
f ◦ g −1 dPg .
Pg (A) g−1 (A)
Pg (A) A
Proposition 2.7. E (E(f |(g, A))) = E(f ).
Proof.
Z
E(f |(g, A)) dP
Z Z
1
=
f dP dP
Pg (A) X g−1 (A)
Z
Z
1
f dP
=
dP
Pg (A) g−1 (A)
X
E(E(f |(g, A))) =
X
= E(f ).
In general we can define the conditional expectation of a random variable f
with respect to a sub σ-algebra M of Σ as a function E(f |M) : X → Rn such that
(i) E(f |M)|M is a random variable;
R
R
(ii) A E(f |M) dP = A f dP
for all A ∈ M. Hence in particular it satisfies E(E(f |M)) = E(f ).
Definition 2.8. A discrete time stochastic process is a sequence of pairs {(fn , Σn )}
where fn is a real-valued random variable on (X, Σn , P ) such that Σn ⊆ Σn+1 ⊆ Σ for all
n.
A discrete time martingale is a stochastic process for which
(i) E(|fn |) < ∞ for all n;
(ii) E(fn+1 |Σn ) = fn .
A discrete time Markov chain is a stochastic process for which
−1
(B)C )
P (fn−1 (B)|Σm ) = P (fn−1 (B)|fm
for m ≤ n and all B ∈ BRn . We can modify the above notions for continuous time
stochastic processes with uncountable indexing sets.
9
3
Analysis on Other Structures
We will study limits on the following spaces:
ks
outer
o measure
oo
ooo
o
o
oo
s{ oo
topological ks
measurable ks
measureKS space
ks
uniform
KS
metric ks
KS
onormed
ooo
o
o
oo
ooo
o
o
o
ks
topological
groups oooooo
Banach
=E
KS
o
o
ooo
ooo
o
o
s{
topological
vector
KS
Riesz ks
Banach lattice ks
ks
inner product
KS
Hilbert
KS
Rn , Cn
We’ve already described how topological spaces induce measurable spaces via the Borel
σ-algebra, which closes the topology under complementation. Outer measure spaces are
simply measurable spaces with an outer measure. Recall from Carathéodory that any
outer measure can be restricted to a sub σ-algebra on which it is a measure.
3.1
Topological and Uniform Spaces
Definition 3.1. Let {xn } be a sequence of points in a topological space. We say {xn }
converges to a point x ∈ X if for every open U 3 x there is some N ∈ N such that
N ≥ n ⇒ xn ∈ U . A topological space is sequentially compact if every sequence has
a convergent subsequence.
Proposition 3.2. If X is a first countable topological space and A ⊂ X, then x ∈ A iff
there is a sequence xn → x.
Definition 3.3. A directed set is a poset D such that for any a, b ∈ D, there is a c
such that a ≤ c and b ≤ c. A net in X is a map φ : D → X. A net is eventually in
A ⊂ X if there is some α ∈ D such that φ(β) ∈ A for all β ≥ α. A net converges to x
if for every neighborhood U of x, the net is eventually in U .
Proposition 3.4. A space is Hausdorff iff any two limits of a net (points to which the
net converges) are equivalent.
Definition 3.5. A uniform space is a pair (X, Υ) where Υ is a collection of subsets
of X × X, called a uniformity, such that
(i) If E ∈ Υ and E ⊆ F , then F ∈ Υ,
(ii) E, F ∈ Υ ⇒ E ∩ F ∈ Υ,
(iii) ∀E ∈ Υ, ∆ ⊆ E where ∆ = {(x, x) : x ∈ X},
(iv) If E ∈ Υ, then there is an F ∈ Υ such that (x, y), (y, z) ∈ F ⇒ (x, z) ∈ E,
10
(v) ∀E ∈ Υ, E −1 = {(y, x) : (x, y) ∈ E} ∈ Υ.
Elements of Υ are called entourages. A map f : (X, Υ1 ) → (Y, Υ2 ) between uniform
spaces is uniformly continuous if for every F ∈ Υ2 there is an enourage E ∈ Υ1 such
that (x, y) ∈ E ⇒ (f (x), f (y)) ∈ F .
One can define a subset U of a uniform space (X, Υ) to be open iff for all x ∈ U ,
there is an entourage Ex such that x̄ = {y : (x, y) ∈ Ex } ⊂ U . It’s easy to verify that
this collection of open sets generates a topology on (X, Υ). Uniformly continuous maps
between uniform spaces are continuous maps between their induced topological spaces.
Definition 3.6. Let F be a filter on a uniform space (X, Υ). F is a Cauchy filter if
for every entourage E there is a set A ∈ F for which A × A ⊆ E. A filter converges
to a point x ∈ X if its induced net converges to x (the induced net has the filter as the
directed set and inclusions as maps). A uniform space is complete if every Cauchy filter
converges.
Definition 3.7. A topological group is a group which is also a topological space
such that multiplication and inversion are continuous. Hence a topological module
(or vector space) is a topological group with continuous scalar multiplication from a
ring (or field).
The topology of a topological group is hence closed under a unary operation
U −1 = {x−1 : x ∈ U }.
A subset U ⊂ G is symmetric if U −1 = U . A topological group induces a uniform space
by defining a set E ⊆ G × G to be an entourage iff it contains the set {(x, y) : xy −1 ∈ U }
for a neighborhood U of 1.
Definition 3.8. A left (right) Haar measure on G is a nonzero Radon measure µ
on BG such that µ(xU ) = µ(U ) (µ(U x) = µ(U )) for all x ∈ G and U ∈ BG .
Theorem 3.9. Let G be a topological group.
(a) A Radon measure µ is a left Haar measure iff
Z
Z
f dµ =
fy dµ
G
G
for all y ∈ G where f is a nonnegative function with compact support and
fy (x) = f (y −1 x).
(b) If µ is a left Haar measure, then µ(G) < ∞ iff G is compact.
(c) If µ and ν are left Haar measures, then µ = cν for some c > 0.
(d) If µ is a left Haar measure and ν is a right Haar measure, then µ ν.
11
3.2
Metric Spaces
Definition 3.10. A metric space is a pair (X, d) where d : X × X → [0, ∞) satisfies
(i) d(x, y) = 0 ⇔ x = y (positive-definiteness),
(ii) d(x, y) = d(y, x) (symmetry),
(iii) d(x, z) ≤ d(x, y) + d(y, z) (triangle inequality).
The map d is called a metric. A map satisfying only the first two is called a semimetric.
Let x ∈ (X, d). We define the open ball of radius r centered at x as the set
Br (x) = {y : d(x, y) < r}
and the closed ball of radius r centered at x as
Br (x) = {y : d(x, y) ≤ r}.
It is easy to verify that the sets pairs Er = {(x, y) : d(x, y) < r} gives a uniformity on
(X, d) and the set of open balls yields a topology. A map f : (X, d1 ) → (Y, d2 ) is called
an isometry if
d1 (x, y) = d2 (f (x), f (y)).
It’s trivial to verify that an isometry is a uniformly continuous map between the induced
uniform spaces of (X, d1 ) and f ((X, d1 )). In fact, it suffices to require that the map
satisfies
d2 (f (x), f (y)) ≤ c d1 (x, y),
a condition called Lipschitz continuity, to ensure that it is uniformly continuous
(clearly preimages of entourages are also entourages).
Next we will show how a metric space induces an outer measure space and in turn
restricts (via Carathéodory) to a measure called a Hausdorff measure.
Let S ⊆ X. We define the diameter of S as
diam(S) = sup{d(x, y) : x, y ∈ S}.
Now for δ > 0 define
Hδd (S) = inf
(∞
X
diam(Ui )d : S ⊂
i=1
∞
[
)
Ui , diam(Ui ) < δ
.
i=1
We then define
H d (S) = lim Hδd (S).
δ→0
d
One can verify that H is an outer measure on 2X and is in fact a measure when restricted
to the Borel sets. By this process, metric spaces induce measure spaces.
Theorem 3.11. If H d (S) < ∞, then H q (S) = 0 for all q > d. If H d (S) > 0, then
H q (S) = ∞ for all q < d.
Hence there is some number d(A) = sup{p : H p (A) = ∞} = inf{p : H p (A) = 0},
called the Hausdorff dimension of A.
12
Definition 3.12. A sequence {xn } in a metric space is a Cauchy sequence if for
every ε > 0, there is some N such that n, m ≥ N ⇒ d(xn , xm ) < ε. A sequence {xn }
converges to x ∈ X if for every ε > 0, there is an N such that n ≥ N ⇒ d(xn , x) < ε.
A metric space is complete if every Cauchy sequence converges.
A complete metric space induces a complete uniform space. One can construct appropriate definitions of continuity and uniform continuity in metric spaces as well.
Definition 3.13. A metric space (X, d) is totally bounded if for every ε > 0, there is
a finite collection {Bε (xk )} of open balls such that X ⊆ ∪nk=1 Bε (xk ).
Theorem 3.14. A metric space is compact iff it is complete and totally bounded iff it
is sequentially compact.
Proof.
Suppose (X, d) is compact. Open sets in the induced topology are balls
centered at points. Hence since ∪x∈X Bε (x) is an open cover of X, there is a finite
subcover ∪nk=1 Bε (xk ). Hence X is totally bounded. Now let {xn } be a Cauchy sequence
in X. We must have that infinitely many terms of the sequence are contained in some
ball Bε/2 (x) (otherwise we could not have a finite cover given that the terms are getting
closer). Suppose {xn }n≥k ⊂ Bε/2 (x), then for large enough n we have
d(xn , x) ≤ d(xn , xk ) + d(xk , x) <
ε ε
+ = ε.
2 2
Thus xn → x and X is complete.
Suppose (X, d) is complete and totally bounded and let {xn } be a sequence in X. Since
X is totally bounded, infinitely many of these terms must be in a ball of finite radius
r. By induction there must be infinitely many of the terms in a ball of radius r/2n . It
follows that for large enough n, {xn }n≥N is a Cauchy sequence and hence converges by
completeness. Hence {xn } has a convergent subsequence.
Suppose (X, d) is sequentially compact and let {Br (xα )} be an open cover for X.
Suppose there is no ε for which there is a finite subcover {Bε (xn )} of X. Then for any
y1 ∈ X, there is some y2 ∈ X such that d(y1 , y2 ) ≥ ε. Since X 6⊆ Bε (y1 ) ∪ Bε (y2 ),
there must be a y3 such that d(y1 , y3 ) ≥ ε and d(y2 , y3 ) ≥ ε. So by induction and the
assumption that {Br (xα )} is not a finite open cover, we have a sequence {yn } such that
d(yn , ym ) ≥ ε for all n, m and cannot have a convergent subsequence (since convergent
implies Cauchy). This contradicts sequential compactness, so {Br (xα )} must have a finite
subcover.
Now consider the set C(X, Y ) of continuous maps between metric spaces (X, d1 ) and
(Y, d2 ) with (X, d1 ) compact. Let us define the function
d(f, g) = sup{d2 (f (x), g(x))}.
x∈X
d clearly satisfies positive-definiteness and symmetry. We also have that if h ∈ C(X, Y ),
then
d(f, g) = sup{d2 (f (x), g(x))}
x∈X
≤ sup{d2 (f (x), h(x)) + d2 (h(x), g(x))
x∈X
≤ sup{d2 (f (x), h(x))} + sup{d2 (h(x), g(x))}
x∈X
x∈X
= d(f, h) + d(h, g).
13
The fact that (X, d1 ) is compact and the functions are continuous gives us that d(f, g) <
∞ for all f, g ∈ C(X, Y ). Thus (C(X, Y ), d) is a metric space.
Definition 3.15. Let (Xn , dn ) be metric spaces. We define the p-metric on the product
Xn1 Xn by
!1/p
n
X
dp ((x1 , ..., xn ), (y1 , ..., yn )) =
di (xi , yi )p
i=1
and
d∞ ((x1 , ..., xn ), (y1 , ..., yn )) = max {di (xi , yi )}.
1≤i≤n
The corresponding metric space is called the product metric space of degree p.
Let (X, d) be a metric space and P (x, y) be the set of continuous functions from [0, 1]
to X as metric spaces such that γ(0) = x and γ(1) = y for all γ ∈ P (x, y). Elements of
this set are called paths from x to y. A metric space is path-connected if for any two
points x, y ∈ X, there is a path from x to y.
3.3
Normed Spaces
Definition 3.16. Let F be a field. A valuation on F is a map | · | : F → [0, ∞) such
that
(i) |x| = 0 ⇔ x = 0,
(ii) |xy| = |x||y|,
(iii) |x + y| ≤ |x| + |y|.
We call (F, | · |) a valuation field.
We have that |1| = |1|2 ⇒ |1| = 1. Moreover
|1| = |(−1)2 | = | − 1|2 ⇒ | − 1| = 1.
Also for x 6= 0
|1| = |x−1 x| = |x−1 ||x| ⇒ |x−1 | = |x|−1 .
A valuation field induces a metric space with metric
d(x, y) = |x − y|.
Definition 3.17. A normed space over a valuation field (F, | · |) is a pair (X, k · k)
where X is an F -vector space and k · k : X → [0, ∞) such that
(i) kxk = 0 ⇔ x = 0,
(ii) kλxk = |λ|kxk for all λ ∈ F ,
(iii) kx + yk ≤ kxk + kyk.
14
Certainly every valuation field is a normed space over itself. And every normed space
induces a metric space with metric d(x, y) = kx − yk. Condition (iii) above also gives us
that
|kxk − kyk| ≤ kx + yk.
We define unit ball of a normed space as the set
U (X) = {x ∈ X : kxk ≤ 1}.
Definition 3.18. A normed space is a Banach space if its induced metric space is
complete.
Let L(X, Y ) denote the set of homomorphisms between two F -normed spaces. Then
L(X, Y ) is an F -normed space where we define (λf )(x) = λf (x) and (f + g)(x) =
f (x) + g(x). The norm, called the operator norm, is defined by
kf k = sup kf (x)k.
kxk=1
Elements f ∈ L(X, Y ) are called linear maps. A linear map is bounded if kf k < ∞
and unbounded otherwise.
Proposition 3.19. Let f ∈ L(X, Y ) where X, Y are C-normed spaces. Then
(a) kf (x)k ≤ kf kkxk.
(b) f is continuous iff it is bounded.
Proof. (a) This is trivial when x = 0. For any nonzero x ∈ X, we can write x = λu
1
x, which
with kuk = 1. This is possible since for any x, one can consider the vector kxk
1
has norm 1. Hence we have λ = kxk and u = kxk x.
sup
x6=0
kf (λu)k
|λ|kf (u)k
kf (x)k
kf (x)k
= sup
= sup
= sup
= sup kf (x)k = kf k.
kxk
kλuk
|λ|kuk
x6=0
x6=0
kxk=1 kxk
kxk=1
(b) If f is continuous then we have that at any point x for every ε > 0 there is a δ > 0
such that
kx − yk < δ ⇒ kf (x) − f (y)k < ε
for any y ∈ X. In particular continuity at 0 gives us that kxk < δ ⇒ kf (x)k < ε. If we
let ε = 1, then there is a δ such that kxk < δ ⇒ kf (x)k < 1. Hence
kf (x)k
1
< .
δ
kxk6=0 kxk
kf k = sup
Now if kf k is bounded, then kf k = M . Hence for any ε > 0, if we chose δ = ε/M ,
then we have
kx − yk < ε/M ⇒ kf (x) − f (y)k = kf (x − y)k ≤ kf kkx − yk < ε.
15
P
Definition 3.20.
Let
{x
}
be
a
sequence
in
X.
We
say
that
the
series
n
n∈N xn conP
P
verges to x if n xn → x. The series is absolutely convergent if n∈N kxn k < ∞.
Theorem 3.21. A normed space is Banach iff every absolutely convergent series converges.
P
P
Proof. Suppose X is Banach and n kxn k < ∞. Let Sk = k1 xn and k ≥ m, then
we have
k
X
kSk − Sm k ≤
kxn k → 0
m+1
as k, m → ∞ since the series is absolutely convergent. But this implies that Sk is Cauchy
and hence converges. Thus the series converges.
Now suppose that every absolutely convergent series converges and {xn } be a Cauchy
sequence. Let {nj } be a sequence such that n, m ≥ nj ⇒ kxn − xm k < 2−j . Now let
P
y1 = xn1 and yj = xnj − xnj−1 . Then k1 yj = xnk and
∞
X
kyj k ≤ ky1 k +
1
∞
X
2−j = ky1 k + 1 < ∞.
1
Thus {xnk } → x for some x ∈ X. But since {xn } is Cauchy, it must have the same limit
as {xnk }.
3.4
The Bochner Integral
Here we define the notion of the integral of a measurable function f defined on a measure
space that takes values in a Banach space B. We similarly define a simple function as
one of the form
n
X
φ(x) =
ci χAi (x)
i=1
where this time ci is a vector in our Banach space. We define the integral of this simple
function intuitively:
Z
n
X
φ(x) dµ =
ci µ(Ai ).
X
i=1
Definition 3.22. Let f : (X, Σ, µ) → B be a measurable function. f is Bochner
integrable if there is a sequence {sn } of simple functions such that
Z
lim
kf − sn k dµ = 0.
n→∞
X
If f is Bochner integrable, we define its Bochner integral as
Z
Z
f dµ = lim
sn dµ
X
n→∞
X
where the limit is taken in the norm topology on B.
Many of the same theorems of the general integral extend to the Bochner integral,
including dominated convergence.
16
3.5
Lp Spaces
We’ll now explore a special example of Banach spaces. We previously explored the space
L1 (X, µ) of Lebesgue integrable functions on X. Let f : X → C be a measurable function
and define
Z
1/p
p
kf kp =
|f | dµ
.
X
We define the corresponding set,
Lp (X, µ) = {f : X → C : kf kp < ∞}.
Lemma 3.23. Let a, b, λ ∈ R such that 0 ≤ a, b and 0 < λ < 1, then
aλ b1−λ ≤ λa + (1 − λ)b,
with equality iff a = b.
Proof.
If b = 0, then it holds. Otherwise let t = a/b, then we have to show
t ≤ λt + (1 − λ). We can find by calculus that tλ − λt attains its maximum at t = 1
since it is strictly decreasing for t > 1 and strictly increasing for t < 1, and its maximum
is precisely 1 − λ.
λ
Theorem 3.24. (Hölder’s Inequality) Suppose 1 < p < ∞ and p−1 + q −1 = 1. If
f ∈ Lp and g ∈ Lq are measurable functions on X, then
kf gk1 ≤ kf kp kgkq .
Proof. If kf kp = 0 or kgkq = 0, then it holds since we must have that f = 0 a.e. or
g = 0 a.e. Similarly it is trivial if kf kp = ∞ or kgkq = ∞. Note that kλf kp = |λ|kf kp
for some constant λ. If kf kp = λ it can be normalized by simply factoring out 1/λ. Thus
we can assume that kf kp = kgkq = 1. Letting a = |f |p , b = |g|q , λ = p−1 , and using the
above lemma, we obtain
|f ||g|q(1−p
−1 )
≤ p−1 |f |p + (1 − p−1 )|g|q .
The requirement that p−1 + q −1 = 1 is equivalent to q = p/(p − 1). Thus we have
|f g| = |f ||g| ≤ p−1 |f |p + q −1 |g|q .
Integrating both sides, we obtain
Z
Z
p
−1
−1
kf gk1 ≤ p
|f | + q
|g|q = p−1 + q −1 = 1 = kf kp kgkq .
Proposition 3.25. If f, g ∈ Lp , then f + g ∈ Lp .
Proof. We have |f + g|p ≤ (2 max{|f |, |g|})p ≤ 2p (|f |p + |g|p ). The integral of the
right hand side is finite.
17
Theorem 3.26. (Minkowski’s Inequality) If 1 ≤ p < ∞ and f, g ∈ Lp , then
kf + gkp ≤ kf kp + kgkp .
Proof. If p = 1 or f + g = 0 a.e., then this holds. Otherwise we have
|f + g|p = |f + g||f + g|p−1 ≤ (|f | + |g|)|f + g|p−1 .
If we let q be such that p−1 + q −1 = 1, then |f | + |g| ∈ Lp and |f + g|p−1 ∈ Lq since
|f + g| ∈ Lp by the previous proposition. Applying Hölder’s Inequality gives us
Z
|f + g|p ≤ kf kp k|f + g|p−1 kq + kgkp k|f + g|p−1 kq
Z
1/q
p
= (kf kp + kgkp )
|f + g|
.
Thus
Z
kf + gkp =
p
1−(1/q)
≤ kf kp + kgkp .
|f + g|
It follows that (L (X, µ), k · kp ) is a seminormed space. It’s not quite a normed
space since we may have any f whose support set has Lebesgue measure 0 for which
kf kp = 0. However, the subspace generated by all differences f − g such that f = g
a.e. is a seminormed subspace. Quotienting out gives us a normed space Lp (X, µ) =
Lp (X, µ)/hf − g : f = g a.e.i.
p
Theorem 3.27. For 1 ≤ p < ∞, Lp (X, µ) is a Banach space.
Proof. We need only show that the space is complete. Recall from 3.21 that a space
is Banach iff every absolutely convergent series is convergent. SoP
suppose {fk } ⊂ Lp is
a sequence of functions
which is absolutely
convergent. That is, ∞
k=1 kfk kp = C < ∞.
P∞
Pn
Now let Gn = k=1 |fk | and G = k=1 |fk |. Then
kGn kp =
n
X
k=1
|fk |
≤
p
n
X
kfk kp ≤ C
k=1
for all n. Thus by the monotone convergence theorem we have
Z
Z
p
G = lim Gpn ≤ C p .
P
So GP
∈ Lp , and hence ∞
is in Lp . If
k=1 fk converges. We wish to show that its limit
P
F = ∞
∈ Lp . It remains toP
show that
fk → F in the
k=1 fk , then |F | ≤ G and hence F
P
n
n
p
p
p
L norm.
∈ L1 . So
R fk | ≤
Pn Notep that1 since |F | ≤ G and k=1 fk ≤ |F |, |F − k=1
P(2G)
n
p
|F − k=1 fk | ∈ L . Thus by dominated convergence we have |F − 1 fk | → 0. Definition 3.28. If f is a measurable function on X, then we define the essential
norm of f as
kf k∞ = inf{a ≥ 0 : µ({x : |f (x)| > a}) = 0}.
18
(Note this is a slight modification to essential supremum, which is not a norm). We also
define
L∞ (X, M, µ) = {f : X → C : kf k∞ < ∞}.
By similarly letting I denote the set of equivalence classes of functions such that f = g
iff f = g a.e., we define the L∞ space as L∞ = L∞ /I.
Theorem 3.29. L∞ is a Banach space.
Proof. Suppose f, g ∈ L∞ . Then it is easy to see that kf +gk∞ ≤ kf k∞ +kgk∞ < ∞.
So f + g ∈ L∞ . Furthermore it is easy to see that kλf k∞ = |λ|kf k∞ < ∞ for a scalar
λ. So λf ∈ L∞ . Hence L∞ is a vector space. It’s also trivial that kf k∞ ≥ 0 and that
kf k∞ = 0 iff f = 0 a.e. So L∞ is a normed space.
Now let {fn } ⊂ L∞ be a Cauchy sequence. Cauchy sequences in L∞ are Cauchy in
measure, so fn → f a.e. Suppose kf k∞ = ∞. Then
kf − fn k∞ ≥ |kf k∞ − kfn k∞ | = ∞.
This contradicts the convergence to f . So f ∈ L∞ .
Proposition 3.30. Suppose 1 < p < q < r ≤ ∞. Then Lq ⊂ Lp + Lr , Lp ∩ Lr ⊂ Lq ,
and kf kq ≤ kf kλp kf k1−λ
where λ ∈ (0, 1) is defined by
r
q −1 = λp−1 + (1 − λ)r−1 , that is, λ =
3.6
q −1 − r−1
.
p−1 − r−1
Inner Product Spaces
Definition 3.31. Let X be a C-vector space. An inner product on X is a map
h·, ·i : X × X → C satisfying
(i) hax + by, zi = ahx, zi + bhb, zi (first component linearity),
(ii) hx, yi = hy, xi (conjugate symmetry)
(iii) hx, xi ≥ 0 and hx, xi = 0 ⇔ x = 0 (positive definitness).
(X, h·, ·i) is called an inner product space.
(i) and (ii) above imply that
hx, ay + bzi = āhx, yi + b̄hx, zi,
making h·, ·i a sesquilinear form. Furthermore, inner product spaces induce normed spaces
by defining
p
kxk = hx, xi.
A Hilbert space is an inner product space whose induced normed space is Banach.
Proposition 3.32. (Schwarz Inequality) |hx, yi| ≤ kxkkyk.
19
Proof. If hx, yi = 0, then it holds. Otherwise let α = sgnhx, yi and z = αy. Then
hx, zi = hz, xi = |hx, yi| .
Hence for any t ∈ R we have
0 ≤ hx − tz, x − tzi = kxk2 − 2t |hx, yi| + t2 kyk2 .
This value takes a minimum at t = kyk−2 |hx, yi|, and substituting this for t gives us
0 ≤ kxk2 − kyk−2 |hx, yi|2 .
Two elements x, y ∈ X are orthogonal if hx, yi = 0. Let A be a subspace of X, then
we define the orthogonal complement of A by
A⊥ = {y ∈ X : hx, yi = 0 for all x ∈ A}.
A subset {xα } of an inner product space is orthonormal provided that kxα k = 1 and
hxα , xβ i = 0 for all α 6= β.
Theorem 3.33. Let H be a Hilbert space, M be a closed subspace of H.
(a) H = M ⊕ M ⊥ .
(b) H has an orthonormal basis
Definition 3.34. Let X, Y be inner product spaces and L(X, Y ) be the linear maps
between them. f ∈ L(X, Y ) and g ∈ L(Y, X) are adjoints if
hf (x), yi = hx, g(y)i
for all x ∈ X and y ∈ Y . The adjoint g of f is often denoted f ∗ . f is unitary if
hx, yi = hf (x), f (y)i
for x, y ∈ X. f is compact if for any bounded A ⊆ X, f (A) is relatively compact.
If f ∈ L(X, X) = L(X), then f is Hermitian if f = f ∗ . f is normal if kf (x)k =
∗
kf (x)k.
3.7
Riesz Spaces
Recall a lattice is a structure (L, ∨, ∧) which is commutative and associative with respect
to each of the operations and satisfies the following:
(i) x ∨ x = x ∧ x = x (idempotency),
(ii) x ∨ (x ∧ y) = x ∧ (x ∨ y) = x (absorption).
In our case, we will deal with the operations inf and sup on a partially ordered set.
A poset in which any two elements have an infimum and supremum will be called an
ordered lattice.
Definition 3.35. A Riesz space is a vector space which is also an ordered lattice. A
Banach (Hilbert) lattice is a Riesz space which is also a Banach (Hilbert) space.
20
4
Dual Spaces and Operator Algebras
In this section we will consider maps between various spaces X and Y in the special cases
when Y = X and when Y = F , the underlying field. Such maps will respectively be
called operators and functionals.
4.1
Dual Spaces
Lemma 4.1. Let ν and µ be finite measures on (X, Σ). Then either ν ⊥ µ or there is
some ε > 0 and E ∈ Σ such that µ(E) > 0 and ν ≥ εµ on E.
Proof. Let X = Pn t Nn be the Hahn decomposition of ν − n−1 µ, P = ∪n Pn , and
N = ∪n Nn . Then we have 0 ≤ ν ≤ n−1 µ on N for all n and thus that ν(N ) = 0. Now
if µ(P ) = 0, then 0 ≤ ν − n−1 µ = ν on P and ν ⊥ µ. Otherwise µ(P ) > 0 and we have
µ(Pn ) > 0 for some n and hence ν ≥ n−1 µ > 0 with ε = n−1 .
Theorem 4.2. (Lebesgue-Radon-Nikodym) Let ν be a σ-finite signed measure
and µ be a σ-finite positive measures on (X, Σ). Then there exist unique σ-finite signed
measures λ and ρ such that λ ⊥ µ, ρ µ, and ν = λ+ρ. Moreover there is a µ-integrable
function f such that
Z
ρ(A) =
f dµ.
A
Here f is called the Radon-Nikodym derivative, and we write f =
dρ
.
dµ
Proof. First suppose ν and µ are finite measures. Let
Z
+
f dµ ≤ ν(E) for all E ∈ Σ}.
F = {f ∈ L :
E
0 ∈ F , so F is nonempty. Moreover if f, g ∈ F , then h(x) = max{f (x), g(x)} ∈ F since
if A = {x : f > g}, then
Z
Z
Z
h dµ =
f dµ +
g dµ ≤ ν(E ∩ A) + ν(E − A) = ν(E).
E
E∩A
E−A
R
R
Let a = supf ∈F { f dµ} and choose a sequence {fn } such that fn dµ → a. Let gn =
max{f1 , ..., fn } and f = supn fn . gn ∈ F , gn ≥ fn , and gn → f pointwise (otherwise we
must have that fn > f Ron some setR of nonzero measure, which contradicts the definition
of f ). Hence
R we have gn dµ → f dµ. Thus by the monotone convergence theorem,
f ∈ F and f dµ = a.
R
Now consider the measure λ = ν − f dµ. Since f ∈ F , we have λ is positive.
Moreover from the previous
lemma, either λ ⊥ µ or there
is a set E for which µ(E) > 0
R
R
and εµ ≤ λ = ν − f dµ on E. But this implies E (εχE + f ) dµ ≤ ν(E) and hence
f + εχE ∈ F . But then we have
Z
(εχE + f ) dµ = εµ(E) + a > a,
E
which contradicts the definition of a, so we must have λ ⊥ µ. By defining ρ =
clearly have ρ µ and ν = λ + ρ. I leave uniqueness to the reader.
21
R
f dµ, we
Now suppose ν and µ are strictly σ-finite measures. Then X is a countable disjoint
union of µ-finite sets and a countable disjoint union of ν-finite sets. We apply the previous results and take limits.
Lastly if ν is a signed measure, then we repeat the above for ν + and ν − .
Proposition 4.3. Let ν be a σ-finite signed measure and µ, λ be σ-finite measures such
that ν µ λ. Then
(a) If g ∈ L1 (X, ν), then g ◦
dν
dµ
∈ L1 (X, µ) and
Z
Z
dν
dµ.
g dν = g ◦
dµ
(b)
dν
dν dµ
=
λ-a.e.
dλ
dµ dλ
Proof. Let us suppose ν is positive and then apply the results to ν + and ν − . Suppose
g = χE , then (a) reads
Z
Z
dν
dµ,
χE dν = χE
dµ
which reduces to
Z
Z
dν
dν =
ν(E) =
, dµ,
E dµ
E
which is true by definition of the Radon-Nikodym derivative. It follows that (a) is true for
simple function and hence measurable functions using monotone convergence. Applying
to positive and negative parts of the measure makes it hold for functions in L1 (X, ν).
(b) If we set g = χE (dν/dµ), then we obtain
Z
Z
dν
dν dµ
dµ =
dλ
ν(E) =
E dµ dλ
E dµ
for all E ∈ Σ, which gives the result.
One can think of a measure as a functional on a σ-algebra. In that case, the LebesgueRadon-Nikodym theorem gives us a universal representation of a measure with respect
to another measure and some element (A, f ) ∈ Σ × L1 (X, µ). In this sense we have a
representation
Z
ν(A) = hA, f iµ =
f dµ.
A
Let us now turn to the case when X is a Hausdorff space and where we let C(X)
denote the set of continuous functions f : X → C. We will consider the uniform norm
defined on such functions by
kf k∞ = sup{|f (x)|}.
x∈X
The induced topology is called the uniform topology.
Definition 4.4. Let F ⊆ C(X). F is equicontinuous at x ∈ X if ∀ε > 0 there is a
neighborhood U 3 x such that y ∈ U ⇒ |f (x) − f (y)| < ε for all f ∈ F . F is pointwise
bounded if for every x, the set {f (x) : f ∈ F } is bounded.
22
Theorem 4.5. (Arzelà-Ascoli) Let X be compact Hausdorff space. Then F ⊆
C(X) is equicontinuous and pointwise bounded iff it is relatively compact in the uniform
topology.
Proof.
Suppose F is equicontinuous and pointwise bounded. It suffices to show
that F is totally bounded, then its closure will be totally bounded and complete, and
hence compact. Then choose ε and let Ux be a neighborhood of x such that if y ∈ Ux
then |f (x) − f (y)| < ε/4 and all f ∈ F . Since X is compact, it can covered by n balls
of radius ε/4 centered at points A = {x1 , ..., xn }. Then since F is pointwise bounded,
{f (xj ) : 1 ≤ j ≤ n} is bounded in C. Hence consider balls of radius ε/4 centered at the
points {f (xj )} and a set B = {z1 , ..., zm } with m ≥ n such that each ball has some zk in
it. The set B A of functions from A to B is finite, and for each φ ∈ B A define
Fφ = {f ∈ F : |f (xj ) − φ(xj )| < ε/4 with 1 ≤ j ≤ n}.
Then certainly F = ∪φ∈B A Fφ . Now pick some f, g in each Fφ , then we have
|f (x) − g(x)| ≤ |f (x) − f (xj )| + |f (xj ) − g(xj )| + |g(xj ) − g(x)|
ε
ε
< + |f (xj ) − φ(xj )| + |φ(xj ) − g(xj )| +
4
4
ε ε ε
< + +
2 4 4
= ε.
Hence F is totally bounded.
Now suppose F is relatively compact, then its closure is complete and totally bounded,
which means F is totally bounded. Hence supx∈X |f (x) − g(x)| ≤ M for all f, g ∈ C(X).
In particular if g = 0 we have supx∈X |f (x)| ≤ M , which gives us pointwise boundedness.
Now ..... |f (x) − f (y)| < ε for all f ∈ F .
A subset A of C(X) separates points if for every x, y ∈ X such that x 6= y, there is
an f ∈ A such that f (x) 6= f (y).
Theorem 4.6. (Stone-Weierstrass) Let X be a compact Hausdorff space. If A
is a closed subalgebra of C(X) under addition, pointwise multiplication, and conjugation which separates points, then either A = C(X) or A = {f ∈ C(X) : f (x0 ) =
0 for some x0 ∈ X}.
Let Cc (X) denote the continuous functions f : X → R for which supp(f ) is compact and C0 (X) = Cc (X). This space is clearly a vector space since supp(f + g) ⊆
supp max{f, g}. The set on the right is certainly compact, and a closed subset of a
compact set is compact. A functional I on this space is positive if f ≥ 0 ⇒ I(f ) ≥ 0.
Theorem 4.7. (Riesz Representation Theorem) Let I be a positive functional on
Cc (X), then there is a unique Radon measure µ such that
Z
I(f ) = f dµ
for all f ∈ Cc (X).
23
In this case we write I(f ) = hf, µi as the pairing. Perhaps not too much of a stretch is
the fact that C0 (X)∗ = M (X) where M (X) is the space of complex Radon measures on
X. That is, every
R functional on C0 (X) coincides with a Radon measure, so we may write
I(f ) = Iµ (f ) = f dµ. Moreover if we define the measure norm
kµk = |µ|(X),
then the above isomorphism is in fact an isometry under the induced metrics of the
uniform and measure norms. This translates to saying
Z
Z
sup
f dµ −
f dν = sup |Iµ (f ) − Iν (f )|
f ∈C0 (X)
X
X
f ∈C0 (X)
= d(Iµ , Iν )
= kµ − νk
= |µ − ν|(X)
= µ+ (X) + µ− (X) + ν + (X) + ν − (X).
We can define derivatives more explicitly in special cases of analytic structures:
Definition 4.8. Let γ be a path in a metric space. We define the its velocity as the
path
d(γ(t + ε), γ(t))
γ 0 (t) = lim
ε→0
ε
if it exists.
Definition 4.9. Let X and Y be locally convex topological vector spaces and f : X → Y
be a map between them. We define the Gâteaux derivative of f in the direction of
u ∈ X as the map Du f : X → Y defined by
f (x + tu) − f (x)
t→0
t
Du f (x) = lim
where the limit is taken with respect to the topology.
Of course if f is a homomorphism we have Du f (x) = f (u).
Definition 4.10. Let X and Y be normed spaces and f : X → Y be a map between
them. We define the Fréchet derivative of f as map Df : X → Y which is linear on
maps between X and Y and which satisfies
kf (x + h) − f (x) − khkDf (x)k
=0
h→0
khk
lim
if it exists (where the limit is again taken with respect to the topology).
Returning to dual spaces, we also have:
Theorem 4.11. Let p and q be conjugate exponents (p−1 + q −1 = 1). If 1 < p < ∞,
then for each φ ∈ (Lp )∗ there is a g ∈ Lq such that
Z
φ(f ) = f g
for all f ∈ Lp . Hence (Lp )∗ = Lq . Moreover if µ is σ-finite, then (L1 )∗ = L∞ .
24
Only one of the spaces is actually Hilbert, and this is L2 (X, µ) with inner product
Z
hf, gi = f g dµ,
and hence clearly kf k = kf k2 .
Now let us define a sublinear functional on a vector space as a map f : X → R
such that f (x + y) ≤ f (x) + f (y) and f (λx) = λf (x). Hence seminorms are sublinear
functionals.
Theorem 4.12. (Hahn-Banach) Let X be a real vector space, M be a subspace,
s be sublinear functional on X, and f be a functional on M such that f (x) ≤ s(x) for
all x ∈ M . Then there is an extension F of f which is a functional on X and satisfies
F (x) ≤ p(x) for all x ∈ X.
Proof. We first extend f to a functional g on the cosets in X/M = M + Rx and then
invoke Zorn’s lemma to conclude the extension to X. First note that for y1 , y2 ∈ M we
have
f (y1 ) + f (y2 ) = f (y1 + y2 ) ≤ s(y1 + y2 ) ≤ s(y1 − x) + s(x + y2 )
and hence
f (y1 ) − s(y1 − x) ≤ s(x + y2 ) − f (y2 ).
Thus,
sup {f (y) − s(y − x)} ≤ inf {s(x + y) − f (y)}.
y∈M
y∈M
If we let α be any number in between (or equal to) and define g : M + R → R by
g(y + λx) = f (y) + λα,
then g is linear and g|M = f . We also have that if λ > 0, then
g(y + λx) = λ (f (y/λ) + α) ≤ λ (f (y/λ) + s(x + (y/λ)) − f (y/λ)) = s(y + λx).
We obtain a similar result for λ < 0. Hence g(x) ≤ s(x) for all x ∈ M + R.
The result is easily extended to complex vector spaces.
Corollary 4.13. Define x̂ : X ∗ → C by x̂(f ) = f (x). Then the map x 7→ x̂ is an
isometric embedding of X into X ∗∗ . In particular the image is dense in X ∗∗ .
Definition 4.14. Let X be a normed space. Then we call the topology induced by
the canonical metric d(x, y) = kx − yk the strong topology. Now define a semimetric
d(x, y) = supf ∈X ∗ |f (x) − f (y)|. Correspondingly the induced topology is the weak
topology. We can see a sequence converges weakly iff f (xn ) → f (x) for all f ∈ X ∗ .
Theorem 4.15. (Banach-Alaoglu) Let X be a normed space. Then the closed unit
ball U ∗ = {f ∈ X ∗ : kf k ≤ 1} is compact in X ∗ with respect to the weak topology.
Q
Proof. For each x ∈ X let Dx = {z ∈ C : |z| ≤ kxk} and define D = x∈X Dx .
Each Dx is closed and bounded in C and is thus compact. By Tychonoff’s theorem an
aribtrary product of compact sets is compact, so D is also compact. Now consider the
map φ : U ∗ → D defined by
φ(f ) = (f (x))x∈X .
Since kf k ≤ 1, we have |f (x)| ≤ kxk and hence f (x) ∈ Dx . Both topologies agree with
convergence, so it suffices to show that im φ is a closed subset of D, which would hence
be compact. Suppose (fα ) is a net in U ∗ that converges to some (λx )x∈X . But then the
map defined by g(x) = λx necessarily satisfies |g(x)| ≤ kxk and hence g ∈ U ∗ .
25
4.2
Operator Algebras
Let X be a normed space and L(X) denote the endomorphisms of X. If f, g ∈ L(X)
then we can certainly define the composition product (f g)(x) = f (g(x)). Hence L(X) is
a C-algebra.
Definition 4.16. A complex algebra is a Banach algebra if it is also a Banach space
whose norm satisfies
kxyk ≤ kxkkyk.
A ∗ -algebra is an algebra together with an involution
∗
: X → X that satisfies
(i) x∗∗ = x (idempotency)
(ii) (x + y)∗ = x∗ + y ∗
(iii) (xy)∗ = y ∗ x∗
(iv) (λx)∗ = λ̄x∗ .
A Banach ∗ -algebra is both a Banach space and ∗ -algebra which also satisfies kx∗ k =
kxk. A C∗ -algebra is a Banach ∗ -algebra that satisfies the C∗ axiom: kx∗ xk = kxk2 .
Example 4.17. Consider the collection of reflexive normed spaces under direct sum and
tensor product and with dual space as the involution and where kAk = dim(A). Normed
∗
-algebra? Yes if we restrict to finite dimensional spaces, but for arbitrary spaces we have
A∗ ⊗ B ∗ ⊆ (A ⊗ B)∗ and dim(A) ≤ dim(A∗ ).
4.3
Spectral Theory
Definition 4.18. Let {Hx }x∈X be a collection of Hilbert spaces such that (X, Σ, µ) is a
measure space. Then we define the direct integral of {Hx }x∈X as
)
(
Z
Z ⊕
M
|s(x)|2 dµ < ∞ /hs − t : s = t µ-a.e.i
Hx dµ = s ∈
Hx :
X
x∈X
X
where s(x) = sx (the xth component of s). This space is a Hilbert space with inner
product defined by
Z
hs, ti =
hs(x), t(x)i dµ.
X
Now if {fx }x∈X is a collection of bounded
operators
R ⊕ in the sense that fx ∈ L(Hx ) for
P
all x and ess supx∈X kfx k < ∞, then x∈X fx ∈ L( X Hx dµ) and we define
!
X
X
fx (s) =
fx (sx ) = (fx (sx ))x .
x∈X
x∈X
R⊕
for s ∈ X Hx .
A projection on a vector space is a map f ∈ L(V ) such that f 2 = f .
Definition 4.19. A projection-valued measure (sometimes called a spectral measure) on a measurable space (X, Σ) is a map π : Σ → L(H) for a Hilbert space H such
that
26
(i) π(A) is a self-adjoint projection for all A ∈ Σ,
(ii) π(X) = 1.
One may also have an induced complex measure µu,v : Σ → C defined by µu,v (A) =
hπ(A)(u), vi.
Theorem 4.20. Let f be a bounded measurable function on (X, Σ, µ) and H be a
Hilbert space, then there exists a unique bounded linear map g ∈ L(H) such that
Z
hg(u), vi =
f dµu,v
X
for all u, v ∈ H.
Let f ∈ L(H). We define an eigenvalue of f to be any λ ∈ C such that f (u) = λu
for some u ∈ H (called an eigenvector of λ). If f is self-adjoint, then its eigenvalues are
real since
λhu, f (u)i = hλu, f (u)i = hf (u), λui = hu, λf (u)i = λ̄hu, f (u)i
which implies λ = λ̄.
Definition 4.21. Let f ∈ L(H) be self-adjoint. We define the spectrum of f as the
set
σ(f ) = {λ ∈ C : f − λ · 1 is not invertible}.
Hence this set includes the eigenvalues of f . Now let us define the operator P[λ,∞) (f )
with λ ∈ R as the projection from H onto supp(f − λ · 1). Let us also define P[λ,κ) (f ) =
P[κ,∞) (f ) − P[λ,∞) (f ) when λ ≤ κ. The set {P[λ,∞) (f )}λ∈R is called the spectral resolution of f . One can verify that this is a projection-valued measure on BR . We furthermore
define a partial ordering on these projections for a given f with PA ≤ PB ⇔ im PA ≤
im PB .
Theorem 4.22. (Spectral Theorem) Let f ∈ L(H) be bounded and self-adjoint.
Then
Zkf k
λ dPλ .
f=
−kf k
For unbounded self-adjoint operators, we have
Z
f=
λ dPλ ,
R
and for normal operators
Z
f=
λ dPλ .
σ(f )
27
5
Analysis on Manifolds
Recall a real (complex) n-manifold is locally homeomorphic to Rn (Cn ).
Definition 5.1. A real (complex) Banach n-manifold is a Hausdorff space M such
that for all x ∈ M there is a neigborhood Ux 3 x and a map ϕx : U → B, with B a
Banach space satisfying dimR B = n (dimC B = n), which is a homeomorphism.
A
chart on M is a homeomorphism ϕ : U → B where U is an open set in M . An atlas of
M is a cover of M by open sets that have charts.
Obviously every Banach manifold has an atlas, since we can just choose {Ux }x∈M .
5.1
Differentiable Manifolds
Definition 5.2. Let f : M → N be a map between two real manifolds of dimensions
m and n respectively. f is differentiable (C k ) if ρ ◦ f ◦ ϕ−1 : Rm → Rn is differentiable
(C k ) for all charts ρ on f ◦ ϕ−1 (Rm ) where ϕ is a chart on M . On Banach manifolds we
generalize by changing Jacobian derivative to Fréchet derivative.
A map f : M → N is a diffeomorphism (C k -diffeomorphism) if f is a differentiable (C k ) homeomorphism whose inverse is also differentiable (C k ).
k
A real manifold is differentiable (C k ) if ϕB ◦ ϕ−1
A is differentiable (C ) on ϕA (A ∩ B)
for any two neighborhoods A, B with charts such that A ∩ B 6= ∅.
Definition 5.3. A C k map f between manifolds M and N of dimensions m, n respectively is an immersion if rank ρf ϕ−1 = m. f is a submersion if rank ρf ϕ−1 = n. An
embedding of one manifold into another is an immersion which is also a homeomorphism.
Theorem 5.4. (Whitney) Every paracompact, connected, differentiable real nmanifold can be embedded in R2n+1 .
Let f : M → N be a C k map between two smooth manifolds of dimensions m, n
respectively. The points in M for which rank f < n are called critical points of f . All
other points in M are regular points of f . A point p ∈ N for which f −1 (p) contains a
critical point is called a critical value of f .
Theorem 5.5. (Sard) Let M and N be smooth manifolds of dimensions m, n respectively with m ≥ n and f : M → N be a C k map. If k ≥ m − n + 1, then the set of critical
values of f has Lebesgue measure 0.
Definition 5.6. We define C ∞ (M ) as the collection of all smooth maps f : M → R
(treating R as a manifold), which is a commutative algebra under pointwise multiplication. We define a derivation (or tangent vector) at x ∈ M as a map D : C ∞ (M ) → R
that satisfies
D(f g) = D(f )g(x) + f (x)D(g).
The set of all derivations at x, denoted Derx (C ∞ (M )), is a vector space called the
tangent space at x. We write Derx (C ∞ (M )) = Tx (M ) for short.
28
Example 5.7. Let γ : (−1, 1) → M be a path in M such that γ(0) = x and define
D(f ) = (f ◦ γ)0 (0) for f ∈ C ∞ (M ) where γ 0 (t) = (ϕ ◦ γ)0 (t) (which is independent of the
chart). Then clearly
D(f g) = (f g ◦ γ)0 (0)
= (f g)0 (x) · γ 0 (0)
= (f 0 (x)g(x) + f (x)g 0 (x)) γ 0 (0)
= f 0 (x)γ 0 (0)g(x) + f (x)g 0 (x)γ 0 (0)
= (f ◦ γ)0 (0)g(x) + f (x)(g ◦ γ)0 (0)
= D(f )g(x) + f (x)D(g).
Clearly the derivation depends only upon γ 0 (0), so the equivalence classes [γ]x that satisfy
γ1 (0) = x and γ10 (0) = γ20 (0) for all γ1 , γ2 ∈ [γ]x give our tangent vectors.
In fact the above definition and example are equivalent. Given a derivation at x one
can consider ϕ−1
x (−1, 1) as a path where ϕx is a chart for a neighborhood of x (we shift as
∞
necessary using Urysohn’s lemma so that ϕ−1
x (0) = x). Then for f ∈ C (M ) we define
−1 0
D(f ) = (f ◦ ϕx ) (0).
Definition 5.8. We define the tangent bundle of M as T (M ) = ∪x∈M Tx (M ). The
cotangent space of x is defined as Tx (M )∗ , and similarly the cotangent bundle is
defined as T ∗ (M ) = ∪x∈M Tx (M )∗ .
The tangent bundle of a C k manifold is a C k−1 manifold. If f : M → N is a
differentiable map between manifolds, then it induces a map between the tangent bundles:
f ∗ : Tx (M ) → Tx (N ) defined by
(f ∗ (v))(g) = v(g ◦ f )
where g ∈ C ∞ (N ) and v ∈ Tx (M ).
Definition 5.9. A differentiable manifold M is a vector fiber bundle of fiber V where
V is a vector space if there is a differentiable manifold B, called the base, a surjective
differentiable map π : M → B, and a surjective diffeomorphism ρ : U × F → π −1 (U ) such
that π −1 (p) = F and π ◦ ρ(x, F ) = x for all p ∈ B and x ∈ U where U is a neighborhood
of p.
A section of a vector fiber bundle M is a differentiable map s : B → M such that
π ◦ s = 1.
A tangent bundle T (M ) is a vector fiber bundle of fiber Rn where B = M and
π(Tx (M )) = x. A section of T (M ) is called a vector field. Hence for a vector field
X : M → T (M ) we have X(x) ∈ Tx (M ). A vector field is in fact a Lie algebra over R.
We define its operations by defining how its image tangent vectors operate as derivations
on smooth functions f ∈ C ∞ (M ):
(X + Y )(p)(f ) = (X(p) + Y (p))(f )
and the Lie bracket
[X, Y ](p)(f ) = X(p) [Y (p)(f )] − Y (p) [X(p)(f )] .
29
Definition 5.10. An n-manifold M is parallelizable if there are n vector fields
X1 , ..., Xn such that for any point p ∈ M , {X1 (p), ..., Xn (p)} is a basis of Tp (M ).
Proposition 5.11. An n-manifold is parallelizable iff T (M ) is diffeomorphic to M ×Rn .
Recall the exterior algebra of an R-algebra A is defined as
Λ(A) =
∞
M
Λk (A)
k=0
where
Λk (A) = A⊗k /hxσ(1) ⊗ · · · ⊗ xσ(k) − sgn (σ)x1 ⊗ · · · ⊗ xk i
with σ is a permutation and where Λ0 (A) = R. We define a k-form in an n-manifold as
a section of Λk (T ∗ (M )), which in local coordinates via a chart has a representation
X
ω=
fr1 ,...,rk dxr1 ∧ · · · ∧ dxrk
1≤r1 ≤···≤rk ≤n
since as a section it’s a map ω : M → Λk (T ∗ (M )) defined by
X
ω : p 7→
fr1 ,...,rk (p) dxr1 ∧ · · · ∧ dxrk .
1≤r1 ≤···≤rk ≤n
Elements of the exterior algebra of the cotangent bundle, Λ(T ∗ (M )), are called differential forms and have the concatenation (or exterior) product:
(f dxi )(g dxj ) = f g dxi ∧ dxj ,
where dxi ∧ dxi = 0. We can also think of a k-form as a real-valued function of k vector
fields defined by
X
ω(p)(X1 , ..., Xk ) =
fr1 ,...,rk (p) dxr1 (X1 (p)) · · · dxrk (Xk (p)).
1≤r1 ≤···≤rk ≤n
We can define the exterior differential (or de Rham derivative) of a simple k-form
as the (k + 1)-form:
k
X
∂f i
dx ∧ dxr1 ∧ · · · ∧ dxrk
dω =
∂x
i
i=1
and extend by linearity to k-forms and then differential forms. One can verify d2 ω = 0
and that the algebra of differential forms yields a complex whose cohomology is called
the de Rham cohomology of M .
Definition 5.12. A partition of unity on a topological space X is a set {αi }i∈I of
continuous functions αi : X → [0, 1] such that
(i) ∀x ∈ X we have a neighborhood Ux containing x on which all but finitely many
αi are 0;
P
(ii) ∀x ∈ X, i∈I αi (x) = 1.
30
(Compactness is sufficient for the existence of a partition of unity on a manifold)
A connected manifold is orientable if there exists an atlas for which all changes of
charts have positive Jacobian determinant. The relation ω1 ∼ ω2 iff ω1 = f ω2 for some
f > 0 is an equivalence relation with two equivalence classes, called orientations of M .
The integral of a differential form is defined with respect to an atlas {Ui , ϕi }i∈I and
a partition of unity {αi } subordinate to the atlas by
Z
XZ
1
n
ω=
αi (x)fi (x) ◦ ϕ−1
i dx · · · dx .
M
i∈I
ϕ(Ui )
Theorem 5.13. (Stokes’) Let M be an n-dimensional C k , oriented, and compact
manifold with boundary and ω be an (n − 1)-form on M . Then
Z
Z
dω =
ω.
M
∂M
Definition 5.14. Let M be a differentiable manifold and f ∈ C ∞ (M ). An affine
connection (or covariant derivative) on M is a map ∇ : Γ(M ) × Γ(M ) → Γ(M ) such
that
(i) ∇aX+bY Z = a∇X Z + b∇Y Z,
(ii) ∇X (Y + Z) = ∇X Y + ∇X Z,
(iii) ∇f X Y = f ∇X Y ,
(iv) ∇X (f Y ) = X(f )Y + f ∇X Y
where the vector field f Y is defined by (f Y )(p) = f (p)Y (p).
If we let {∂i } be the canonical basis of a tangent space (where ∂i = ∂/∂xi ), then for
a given p we have
∇X(p) (Y (p)) = ∇X(p)i ∂i (Y (p)j ∂j )
= X(p)i ∇∂i (Y (p)j ∂j )
= X(p)i ∂i (Y (p)j )∂j + X(p)i Y (p)j ∇∂i ∂j
= X(p)i ∂i (Y (p)k ) + X(p)i Y (p)j Γkij ∂k
where ∇∂i ∂j = Γkij ∂k since the derivative is vector field. The coefficients Γkij are called
the Christoffel symbols of the connection ∇. The above equation also serves as a
definition of a connection if the Christoffel symbols are given.
We can also write the Lie bracket in terms of the tangent space basis {∂i }:
[X, Y ](p) = X(p)(Y (p)i ) − Y (p)(X(p)i ) ∂i = X(p)j ∂j Y (p)i − Y (p)j ∂j X(p)i ∂i .
Definition 5.15. Let ∇ be a connection. The torsion of the connection is a map
T : Γ(M ) × Γ(M ) → Γ(M ) by
T (X, Y ) = ∇X Y − ∇Y X − [X, Y ].
The curvature of ∇ is a map R : Γ(M )3 → Γ(M ) defined by
R(X, Y, Z) = ∇X ∇Y Z − ∇Y ∇X Z − ∇[X,Y ] Z.
31
The curvature of a connection can also be thought as a function taking two vector
fields to an operator on the Lie algebra of vector fields:
(X, Y ) 7→ ∇X ∇Y − ∇Y ∇X − ∇[X,Y ] = R(X, Y, ·).
From here we will use the notation X i to mean X(p)i . If we evaluate the torsion vector
field at a point and look at its component, we have
T (X, Y )(p) = X i Y j Γkij ∂k − Y i X j Γkij ∂k − [X, Y ]
= (Γkij − Γkji )X i Y j ∂k − [X i ∂i , Y j ∂j ]
= (Γkij − Γkji − γijk )X i Y j ∂k
= Tijk X i Y j ∂k
where γijk ∂k = [∂i , ∂j ]. Here we call the constants {Tijk } the torsion components.
Similarly for curvature we have
R(X, Y, Z)(p) = X i ∇∂j (∇Y Z) − Y i ∇∂i (∇X Z) − X i Y j ∇[∂i ,∂j ] Z
= X i Y j Z k ∇∂i ∇∂j ∂k − Y i X j Z k ∇∂i ∇∂j ∂k − X i Y j Z k γijl ∇∂l ∂k
l
= X i Y j Z k ∇∂i Γljk ∂l − Y i X j Z k ∇∂i Γljk ∂l − X k Y j Z i γkj
∇∂l ∂i
i j k l
m
k j i l
m
= X i Y j Z k Γljk Γm
il ∂m − Y X Z Γjk Γil ∂m + X Y Z γkj Γli ∂m
l
m
l m
= X i Y j Z k (Γljk Γm
il − Γik Γjl + γij Γli )∂m
m
= Rijk
X i Y j Z k ∂m .
Proposition 5.16. (Bianchi Identities) We have
X
X
l
l
)
Rijk
=
(∇∂j Tkil − Tjim Tmk
σ(i,j,k)
σ(i,j,k)
and
X
X
m
=
∇∂j Rlki
m
Tikh Rlhj
.
σ(i,j,k)
σ(i,j,k)
We can restrict the notion of the derivative ∇X Y with respect to a vector field to a
path derivative with respect to a curve γ: ∇γ Y . Here we treat a differentiable path γ as
a vector field since γ 0 (t) ∈ Tγ(t) (M ).
Definition 5.17. A vector field X is said to be parallel to a differentiable curve γ if
∇γ X = 0. Conversely a differentiable curve γ is a geodesic if its field of tangent vectors
is parallel along γ.
5.2
Riemannian Manifolds
Definition 5.18. A Riemannian manifold is a manifold where each tangent space
Tp (M ) has an inner product gp , whose direct sum on the tangent bundle is called the
Riemannian metric.
One can verify that the inner product gives a connected Riemannian manifold a metric
space structure, with metric defined by
Z 1
0
d(x, y) = inf
kγ (t)k dt
γxy
0
32
where γxy denotes
a differentiable curve connecting points x and y and k · k is the inp
duced norm g(·, ·). We will use the notation g(X, Y ) for vector fields X, Y to mean
gp (X(p), Y (p)).
Let us define the directional derivative of a function f ∈ C ∞ (M ) with respect to
a vector field X as a vector field defined by
DX f (p) = X(p)(f ).
Definition 5.19. A Riemannian connection on a Riemannian manifold is a connection ∇ satisfying
(i) [X, Y ] = ∇X Y − ∇Y X (torsion free)
(ii) DZ g(X, Y ) = g(∇Z X, Y ) + g(X, ∇Z Y ) (metric).
Theorem 5.20. Every Riemannian manifold has a unique Riemannian connection.
The Christoffel symbols for the connection are given by
Γkij =
1
(∂i glj + ∂j gli − ∂l gij ) g lk
2
where g(X, Y ) = X i Y j g(∂i , ∂j ) = gij X i Y j . We will call the terms gij the Riemannian
components of the manifold.
Definition 5.21. Let us define the map S : Γ(M )4 → C by S(W, X, Y, Z) = g(R(W, X, Y ), Z).
We define the sectional curvature of two vector fields X, Y as σ(X, Y ) = S(X, Y, Y, X) =
g(R(X, Y, Y ), X) when X and Y are orthonormal at p (meaning g(X, X) = g(Y, Y ) = 1
and g(X, Y ) = 0). Otherwise we define it
σ(X, Y ) =
S(X, Y, Y, X)
.
kXk2 kY k2 − g 2 (X, Y )
Definition 5.22. Let g denote the matrix of Riemannian components, |g| be its determinant, f be a continuous real function on a Riemannian manifold M of dimension n,
and {Ω, ϕ} be a chart in which f has compact support. Then we define
Z
Z
p
f dV =
( |g|f ) ◦ ϕ−1 dx1 · · · dxn .
M
ϕ(Ω)
If f has compact support in M , then we define
Z
XZ
f dV =
αi f dV
M
i∈I
M
where {αi }i∈I is a partition of unity subordinate to an atlas {Ωi }i∈I .
33
References
[1] Aliprantis, Charalambos D. and Kim C. Border. Infinite Dimensional Analysis: A
Hitchhiker’s Guide. 3rd Edition. Springer-Verlag. 2006.
[2] Aubin, Theirry. A Course in Differential Geometry. American Mathematical Society.
2001.
[3] Folland, Gerald B. Real Analysis: Modern Techniques and Their Applications. 2nd
Edition. John Wiley & Sons, Inc. 1999.
[4] Lang, Serge. Real and Functional Analysis. 3rd Edition. Springer-Verlag. 1993.
[5] Shiryaev, A.N. Probability. 2nd Edition. Springer-Verlag. 1996.
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