THEOREMS, ETC., FOR MATH 516
Results labeled Theorem Ea.b.c (or Proposition Ea.b.c, etc.) refer to Theorem
c from section a.b of Evans’ book (Partial Differential Equations).
Proposition 1 (=Proposition 10.2). (a) A bounded linear operator is uniformly
continuous.
(b) A linear operator which is continuous at one point is bounded.
Proposition 2 (=Proposition 10.3). The set of all bounded linear operators from
a normed linear space to a Banach space is also a Banach space.
Theorem 3 (=Theorem 10.4, Hahn-Banach). Let X be a vector space, let p be a
real- valued function on X such that (x + y) ≤ p(x) + p(y) and p(αx) = αp(x) for
all x and y in X and all α ∈ R. Suppose f is a linear functional on a subspace S
of X such that f (s) ≤ p(s) for all s ∈ S. Then there is a linear functional F on X
which agrees with f on S such that F (x) ≤ p(x) for all x ∈ X.
Proposition 4 (=Proposition 10.5). Let X be a vector space, let p be a real-valued
function on X such that (x + y) ≤ p(x) + p(y) and p(αx) = αp(x) for all x and y in
X and all α ∈ R. Suppose f is a linear functional on a subspace S of X such that
f (s) ≤ p(s) for all s ∈ S. Let G be an Abelian semigroup of linear operators on X
and suppose that p(Ax) ≤ p(x) for all x ∈ X and f (As) = f (s) for all s ∈ S. Then
there is a linear functional F on X which agrees with f on S such that F (x) ≤ p(x)
and F (Ax) = F (x) for all x ∈ X.
Proposition 5 (=Proposition 10.6). Let X be a normed linear space and let x be
a nonzero element of X. Then there is a bounded linear functional f on X such
that f (x) = kf kkxk.
Proposition 6 (=Proposition 10.7). Let X be a normed linear space, let T be a
subspace of X and let y be an element of X such that d(y, T ) ≥ δ for some positive
δ. Then there is a bounded linear functional f on X such that f (y) = δ, kf k ≤ 1,
and f = 0 on T .
Lemma 7 (=Lemma 10.9). Let X and Y be Banach spaces and let A be a bounded
linear transformation of X onto Y . Then the image of the unit ball of X contains
an open ball of Y with center 0.
Theorem 8 (=Proposition 10.10, Open Mapping Theorem). Let A be a bounded
linear transformation from a Banach space X onto a Banach space Y . Then A is
an open mapping. If A is also one-to-one, then A is an isomorphism.
Proposition 9 (=Proposition 10.11). Let X be a vector space, let k · k1 and k · k2
be two norms on X and suppose that (X, k · k1 ) and (X, k · k2 ) are both Banach
spaces. If there is a constant C such that
kxk1 ≤ Ckxk2
for all x ∈ X, then there is a constant K such that
kxk2 ≤ Kkxk1
1
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THEOREMS, ETC., FOR MATH 516
for all x ∈ X.
Theorem 10 (=Theorem 10.12, Closed Graph Theorem). Let X and Y be Banach spaces and let A : X → Y be a linear transformation. If, whenever hxn i is a
convergent sequence in X with limit x such that hAxn i is a convergent sequence in
Y with limit y, we have y = Ax. Then A is bounded.
Proposition 11 (=Proposition 10.13, Uniform Boundedness Principle). Let X be
a Banach space, let Y be a normed linear space, and let F be a family of linear
operators from X to Y . Suppose that, for every x ∈ X, there is a number Mx such
that kT xk ≤ Mx for all T ∈ F. Then there is a constant M such that kT k ≤ M
for all T ∈ F.
Proposition 12 (=Proposition 8.5). A collection B of subsets of a set X is a base
for a topology on X if and only if
(i) For every x ∈ X, there is B ∈ B such that x ∈ B.
(ii) For every x ∈ X and every B1 and B2 in B such that x ∈ B1 ∩ B2 , there is
B3 ∈ B such that x ∈ B3 ⊂ B1 ∩ B2 .
Proposition 13 (=Proposition 8.6). A topological space satisfies T1 if and only if
points are closed.
Lemma 14 (=Lemma 8.7, Urysohn’s Lemma). Let A and B be disjoint closed
subsets of a normal space X. Then there is a continuous, real-valued function f on
X such that 0 ≤ f ≤ 1 on X, f ≡ 0 on A, and f ≡ 1 on B.
Theorem 15 (=Theorem 8.8, Tietze’s Extension Theorem). Let A be a closed
subset of a normal space X and let f : A → R be continuous. Then there is a
continuous function g : X → R such that g = f on A.
Proposition 16 (=Proposition 10.14). Let X be a vector space.
(a) Let T be a topology on X which makes X a topological vector space. Then there
is a local base B for T such that
(i) If U and V are in B, then there is W ∈ B such that W ⊂ U ∩ V .
(ii) If U ∈ B and x ∈ X, then there is V ∈ B such that x + V ⊂ U .
(iii) If U ∈ B, then there is v ∈ B such that V + V ⊂ U .
(iv) If U ∈ B and x ∈ X, then there is α ∈ R such that x ∈ αU .
(v) If U ∈ B and if α ∈ [−1, 1] is nonzero, then αU ⊂ U and αU ∈ B.
(b) Conversely, if B is a collection of subsets of X which all contain 0 which satisfies
properties (i)–(v), then B is a local base for a topology which makes X into a
topological vector space. This topology is Hausdorff if and only if
T
U = {0}.
(vi)
U ∈B
Proposition 17 (=Proposition 10.15, Tychonoff). Let X be a finite-dimensional,
Hausdorff topological vector space. Then X is isomorphic to Rn for some n.
Proposition 18 (=Proposition 10.16). A subspace of a topological vector space is
closed if and only if it’s weakly closed.
Theorem 19 (=Theorem 10.17, Alaoglu). S ∗ = {f : kf k ≤ 1} is weak∗ compact
in X ∗ .
Lemma 20 (=Lemma 10.18). If K1 and K2 are convex, then so are K1 ∩ K2 , λK1
(for any real λ), and K1 + K2 .
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3
Lemma 21 (=Lemma 10.19). If 0 is an internal point of a convex set K, then the
support function p satisfies the following conditions:
(i) p is positively homogeneous.
(ii) p is subadditive.
(iii) {x : p(x) < 1} ⊂ K ⊂ {x : p(x) ≤ 1}.
Theorem 22 (=Theorem 10.20). If K1 and K2 are disjoint convex set in a vector
space X and if one of them has an internal point, then there is a nonzero linear
functional which separates them.
Proposition 23 (=Proposition 10.21). Let X be a vector space.
(a) Let N be a family of convex subsets of X containing 0 and suppose that they
satisfy the following conditions:
(i) if N ∈ N, then each point of N is an internal point.
(ii) For N1 and N2 in N, there is N3 ∈ N such that N3 ⊂ N1 ∩ N2 .
(iii) If N ∈ N and 0 < |α| ≤ 1, then αN ∈ N.
Then N is a local base for a topology on X which makes X a locally convex topological vector space.
(b) Conversely, if X is a locally convex topological vector space, then there is a local
base N satisfying (i)–(iii).
Proposition 24 (=Proposition 10.22). Let F be a closed convex subset of a locally
convex topological vector space X and let x ∈ X ∼ K. Then there is a continuous
linear functional f on X such that
f (x) < inf{f (y) : y ∈ K}.
Corollary 25 (=Corollary 10.23). A convex subset of a locally convex topological
vector space is strongly closed if and only if it’s weakly closed.
Corollary 26 (=Corollary 10.24). If x and y are distinct points of a locally convex,
Hausdorff topological vector space, then there is a continuous linear functional such
that f (x) 6= f (y).
Lemma 27 (=Lemma 10.25). Let f be a continuous linear functional defined on
a convex subset K of a topological vector space. Then the set of points on which f
attains its maximum is a supporting set of K.
Theorem 28 (=Theorem 10.26, Krein-Milman). Let K be a compact, convex subset
of a locally convex, Hausdorff topological vector space. Then K is the closed convex
hull of its extreme points.
Proposition 29 (=Proposition 9.2, second part). A compact subset of a Hausdorff
space is closed.
Proposition 30 (=Proposition 9.5). A one-to-one continuous function from a
compact space onto a Hausdorff space is a homeomorphism.
Proposition 31 (=Proposition 9.6). The continuous image of a countably compact
space is countably compact.
Proposition 32 (=Proposition 9.7). A space is countably compact if and only if
it has the Bolzano-Weierstrass property.
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THEOREMS, ETC., FOR MATH 516
Proposition 33 (=Proposition 9.8). A sequentially compact space is countably
compact. Every first countable, countably compact space is sequentially compact.
Proposition 34 (=Proposition 9.10). An upper semicontinuous real-valued function on a countably compact space is bounded above and attains its maximum value.
Corollary 35 (=Proposition 9.9). A continuous real-valued function on a countably compact space is bounded above and below, and it attains its maximum and
minimum values.
Proposition 36 (=Proposition 9.11, Dini). Let hfn i be a sequence of upper semicontinuous real-valued functions on a countably compact space X such that hfn (x)i
decreases to 0 for each x ∈ X. Then fn → 0 uniformly on X.
Lemma 37 (=Lemma 9.12). Let A be a collection of subsets of a set X with the
finite intersection property. Then there is a collection B of subsets of X with the
finite intersection property which contains A and which is maximal.
Lemma 38 (=Lemma 9.13). Let B be a maximal collection of subsets of a set X
with the finite intersection property.
(i) Any intersection of finitely many elements of B is an element of B.
(ii) If C ⊂ X and B ∩ C 6= ∅ for all B ∈ B, then C ∈ B.
Theorem 39 (=Theorem 9.14, Tychonoff). Any product of compact spaces is compact.
Proposition 40 (=Proposition 9.15). Let K be a compact subset of a locally compact Hausdorff space X.
(i) Then there is an open set U such that K ⊂ U and U is compact.
(ii) For any such U , there is a continuous nonnegative, real-valued function f
such that f = 0 on Ũ and f = 1 on K. If K is a Gδ set, then this function
also has the property that f < 1 on K̃.
Proposition 41 (=Proposition 9.16). Let {Oλ } be an open cover of a compact subset K of a locally Hausdorff space X. Then there is a finite collection {ϕ1 , . . . , ϕn }
of continuous, nonnegative, real-value functions subordinate to {Oλ } such that
ϕ1 + · · · + ϕn = 1 on K.
Proposition 42 (=Proposition 9.29). Let L be a lattice in C(X) for some compact
Hausdorff space X. If h, defined by
h(x) = inf f (x),
f ∈L
is continuous, then, for any ε > 0, there is a function g ∈ L such that 0 ≤ g −h ≤ ε.
Lemma 43 (=Lemma 9.31). Let L be a family of real-valued functions on a set X
such that
(i) L separates points.
(ii) If f ∈ L and c ∈ R, then f + c and cf are in L.
Then for any a and b in R and any x 6= y in X, there is f ∈ L such that f (x) = a
and f (y) = b.
Lemma 44 (=Lemma 9.32). Let L be a lattice in C(X) for some compact space
X and suppose L satisfies properties (i) and (ii) from Lemma 43. Let p ∈ X and
let F be a closed subset of X with p ∈
/ F . Then for any real numbers a ≤ b, there
is f ∈ L with f ≥ a in X, f > b in K and f (p) = a.
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5
Proposition 45 (=Proposition 9.30). Let L be a lattice in C(X) for some compact
space X and suppose L satisfies properties (i) and (ii) from Lemma 43. Then
L = C(X).
Lemma 46 (=Lemma 9.33). For any ε > 0, there is a polynomial Pε such that
||t| − Pε (t)| < ε for t ∈ [−1, 1].
Theorem 47 (=Theorem 9.34, Stone-Weierstrass). Let X be a compact space and
let A be an algebra in C(X) that contains the constant functions and separates
points. Then A = C(X).
Theorem 48 (=Corollary 9.35, Weierstrass Theorem). Every continuous function
on a closed bounded set of Rn can be uniformly approximated by polynomials.
Proposition 49 (=Proposition 11.17). Let (X, B) be a measurable space, let hµn i
be a sequence of measures on (X, B) that converge setwise to a measure µ, and let
hfn i be a sequence of nonnegative measurable functions that converge pointwise to
a function f . Then
Z
Z
f dµ ≤ lim fn dµn .
Proposition 50 (=Proposition 11.18). Let (X, B) be a measurable space, let hµn i
be a sequence of measures on (X, B) that converge setwise to a measure µ, let
hgn i be a sequence of nonnegative measurable functions that converge pointwise to
a integrable function g, and let hfn i be a sequence of measurable functions that
converge pointwise to a function f . Suppose also that |fn | ≤ g and that
Z
Z
lim gn dµn = g dµ.
Then
Z
lim
Z
fn dµn =
f dµ.
Lemma 51 (=Lemma 11.19). Every measurable subset of a positive set is positive.
The union of countably many positive sets is positive.
Lemma 52 (=Lemma 11.20). Let ν be a signed measure. If E is a set such that
0 < νE < ∞, then there is a positive set A with A ⊂ E and νA > 0.
Theorem 53 (= Proposition 11.21, Hahn Decomposition Theorem). Let ν be a
signed measure on a measurable space (X, B). Then there is a positive set A such
that B = X ∼ A is a negative set.
Lemma 54 (= Proposition 11.10, approximately). Let D be a countable dense
subset of R, let (X, B, µ) be a measure space and suppose that, for every α ∈ D,
there is Bα ∈ B such that µ(Bα ∼ Bβ ) = 0 for any α < β. Then there is a
measurable function f such that f ≤ α on Bα and f ≥ α on B̃α . If g is any other
such function, then g = f a.e.
Theorem 55 (=Theorem 11.23, Radon-Nikodym). Let (X, B, µ) be a σ-finite measure space and let ν be a measure on (X, B) which is absolutely continuous with
respect to µ. Then there is a nonnegative measurable function f such that
Z
νE =
f dµ
E
for any E ∈ B. If g is another such function, then f = g a.e.[µ].
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Proposition 56 (=Proposition 11.24, Lebesgue Decomposition Theorem). Let
(X, B, µ) be a σ-finite measure space and let ν be a σ-finite measure on (X, B).
Then there are two measures ν0 and ν1 such that ν = ν1 + ν2 , ν0 and µ are mutually singular, and ν1 is absolutely continuous with respect to µ. These measures
are unique.
Lemma 57 (=Lemma 11.27). Let (X, B, µ) be a finite measure space, let g ∈ L1 (µ),
let p ∈ [1, ∞), and suppose that there is a constant M such that
Z
gϕ dµ ≤ M kϕkp
for all simple functions ϕ. Then g ∈ Lq (µ) for q = p/(p − 1), and kgkq ≤ M .
Theorem 58 (=Theorem 11.29, Riesz Representation). Let (X, B, µ) be a σ-finite
measure space, let p ∈ [1, ∞), and let F be a bounded linear functional on Lp (µ).
Then there is a unique g ∈ Lq (µ) such that
Z
F (f ) = f g dµ
for all f ∈ Lp (µ).
Theorem 59 (=Theorem 11.30). Let (X, B, µ) be a measure space, let p ∈ (1, ∞),
and let F be a bounded linear functional on Lp (µ). Then there is a unique g ∈ Lq (µ)
such that
Z
F (f ) = f g dµ
for all f ∈ Lp (µ).
Lemma 60 (=Lemma 12.2). Let µ be a measure on A, an algebra of P
sets. If A ∈ A
and hAi i is a sequence of sets in A such that A ⊂ ∪i Ai , then µA ≤ i µAi .
Lemma 61 (=Corollary 12.3, Lemma 12.4, Lemma 12.5). Let µ be a measure on
A, an algebra of sets.
(a) If A ∈ A, then µ∗ A = µA.
(b) µ∗ is an outer measure.
(c) If A ∈ A, then A is µ∗ -measurable.
Proposition 62 (=Proposition 12.6). Let µ be a measure on A, an algebra of sets,
and let E ⊂ X.
(i) For any ε > 0, there is A ∈ Aσ such that E ⊂ A and µ∗ A ≤ µ∗ E + ε.
(ii) There is B ∈ Aσδ such that E ⊂ B and µ∗ E = µ∗ B.
Proposition 63 (=Proposition 12.7). Let µ be a σ-measure on A, an algebra of
sets.
(i) A set E ⊂ X is µ∗ -measurable if and only if there are sets A ∈ Aσδ and B
with µ∗ B = 0 such that E = A ∼ B.
(ii) If µ∗ B = 0, then there is a set C ∈ Aσδ such that B ⊂ C and µ∗ C = 0.
Theorem 64 (=Theorem 12.8, Carathéodory). Let µ be a σ-measure on A, an
algebra of sets. Then µ̄, the restriction of µ∗ to the collection of all µ∗ -measurable
sets, is a measure (on a σ-algebra of sets) which agrees with µ on A. If µ is finite
(or σ-finite), then so is µ̄. If µ is σ-finite, then µ̄ is the only such extension of µ
on the σ-algebra generated by A.
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Proposition 65 (=Proposition 12.9). Let C be a semi-algebra and let µ be a nonnegative set function defined on C with µ∅ = 0 if ∅ ∈ C. Suppose also that
(i) If C ∈ C is the union
P of a finite collection {C1 , . . . , Cn } of disjoints elements
of C, then µC = µCi .
(ii) If C ∈ C is the union
P of a countable collection {C1 , . . . } of disjoints elements
of C, then µC ≤ µCi .
Then µ has a unique extension to a measure on A, the algebra generated by C.
Lemma 66 (=Lemma 12.14). Let {Ri } be a countable collection of disjoint
meaP
surable rectangles with R = ∪Ri a measurable rectangle. Then λ(R) = λ(Ri ).
Lemma 67 (=Lemma 12.15). Let E ∈ Rσδ and x ∈ X. Then Ex is a measurable
function on Y .
Lemma 68 (=Lemma 12.16). Let E ∈ Rσδ with (µ × ν)(E) < ∞. Then g, defined
by g(x) = ν(Ex ), is a measurable function of x and
Z
g dµ = (µ × ν)(E).
Lemma 69 (=Lemma 12.17). If (µ × ν)(E) = 0, then ν(Ex ) = 0 for almost all x.
Proposition 70 (=Proposition 12.18). Let E be a measurable subset of X × Y
with (µ × ν(E) < ∞. Then, for almost all x, Ex is a measurable subset of Y and
g, defined by g(x) = ν(Ex ), is a measurable function of x. Moreover
Z
g dµ = (µ × ν)(E).
Theorem 71 (=Theorem 12.19, Fubini). Let (X, A, µ) and (Y, B, µ) be σ-finite
measure spaces. If f is integrable with respect to µ × ν, then
(i) For almost all x ∈ X, the function fx , defined by fx (y) = f (x, y) is measurable,
(i)’ For almost all y ∈ Y , the function f y , defined by f y (x) = f (x, y) is measurable,
(ii) The function F1 , defined by
Z
F1 (x) =
f (x, y) dν
Y
1
is in L (X),
(ii)’ The function F2 , defined by
Z
F2 (y) =
f (x, y) dµ
X
is in L1 (Y),
R R
R R
R
(iii)
f dν dµ =
f dµ dν =
X
Y
Y
X
f d(µ × ν).
X×Y
Theorem 72 (=Theorem 12.20, Tonelli). Let (X, A, µ) and (Y, B, µ) be σ-finite
measure spaces. If f is nonnegative and measurable with respect to µ × ν, then
(i) For almost all x ∈ X, the function fx , defined by fx (y) = f (x, y) is measurable,
(i)’ For almost all y ∈ Y , the function f y , defined by f y (x) = f (x, y) is measurable,
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THEOREMS, ETC., FOR MATH 516
(ii) The function F1 , defined by
Z
F1 (x) =
f (x, y) dν
Y
is in L1 (X),
(ii)’ The function F2 , defined by
Z
F2 (y) =
f (x, y) dµ
X
is in L1 (Y),
R R
R R
R
(iii)
f dν dµ =
f dµ dν =
X
Y
Y
X
f d(µ × ν).
X×Y
Proposition 73 (=Proposition 12.40). If µ∗ is a Carathéodory outer measure with
respect to Γ, then every function in Γ is µ∗ -measurable.
Proposition 74 (=Proposition 12.41). Let (X, ρ) be a metric space and let µ∗ be
an outer measure on X such that µ∗ A + µ∗ B = µ∗ (A ∪ B) whenever ρ(A, B) > 0.
Then every Borel set is measurable with respect to µ∗ .
Lemma 75 (=Lemma E5.2.1). If u has a weak derivative Dα u for some multiindex α, then this weak derivative is unique.
Theorem 76 (=Theorem E5.2.1). Suppose u and v are in W k,p (Ω) for some nonnegative integer k and some p ∈ [1, ∞].
(i) For any multi-index α with |α| ≤ k, we have Dα u ∈ W k−|α|,p (Ω) and
Dα (Dβ u) = Dβ (Dα u) = Dα+β u if |α| + |β| ≤ k.
(ii) For any real numbers λ and µ, λu + µv ∈ W k,p and Dα (λu + µv) =
λDα u + µDβ v.
(iii) If V is a connected open subset of Ω, then u ∈ W k,p (V ).
(iv) If ζ ∈ Cc∞ (Ω), then ζu ∈ W k,p (Ω) and Di (ζu) = Di ζu + ζDi u.