CMI/BVR
Graduate Analysis 2 HA 1
04-01-2024
From a physiological perspective, the brain must translate the external sources
of energy – sight, sound etc – and encode into electrical patterns which the brain
can understand. The more elaborately we encode information at the moment of
learning, the stronger the memory. The more a learner focuses on the meaning
of the information being presented, the more elaborately he/she will process the
information. When you are trying to drive a piece of information into your brain’s
memory systems, make sure exactly what the information means. Do not try to
memorize the information by rote and pray the meaning will somehow reveal
itself.
John Medina
1. Consider R2 . Plot and understand the ‘shape’ of unit ball B(0, 1) under
the metrics dp . Here 0 stands for (0, 0) ∈ R2 .
2. Show that L∞ is complete. Show that C(X) for compact Hausdorff
with sup metric is complete. Show that the space C(Rn ) with the
metric introduced in class is complete.
Consider (Ω, A, µ) a finite measure space. Let L be
R the collection of
all measurable functions with metric(??) d(f, g) = (|f − g| ∧ 1)dµ is
complete. Show that convergence in this metric is same as convergence
in measure: fn → f iff (∀ > 0)µ(|fn − f | > ) → 0. Show that the
following is also metric and gives the same notion of convergence.
Z
|f − g|
dµ
d(f, g) =
1 + |f − g|
3. (Analysis/Calculus) Show that the function f (x) = exp{−1/x2 } for
x > 0 and f (x) = 0 for x ≤ 0 is a C ∞ -function.
Let a ∈ R. Show that there is a C ∞ -function which is zero on (−∞, a)
and strictly positive on (a, ∞). Show that there is a C ∞ -function which
is strictly positive on (−∞, a) and zero on (a, ∞).
Let a < b ∈ R. Show that there is a C ∞ -function which is strictly
positive on (a, b) and zero outside.
Let a < b ∈ R. Show that there is a C ∞ -function f such that f (x) = 0
for x < a; f (x) = 1 for x > b and in between increases from zero to
one.
Let a < b ∈ R and > 0. Show that there is a C ∞ -function f such
that f (x) = 0 for x < a − ; increases from zero to one during a − to a; f (x) = 1 for a ≤ x ≤ b; then decreases to zero during b to b + ;
f (x) = 0 for x > b + .
1
Given any bounded interval [a, b], show that there is a sequence of C ∞ functions; values in [0, 1], all having compact support; which converges
pointwise to I[a,b] .
Show that the collection of C ∞ -functions with compact support are
dense in Lp (R) for 1 ≤ p < ∞.
By above or otherwise, Show that Lp (R) is separable for 1 ≤ p < ∞.
4. (Ω, A, µ) is a finite measure space and 1 ≤ p < ∞.
Assume Lp is separable show that the following holds:
(♠)there is a sigmafield B ⊂ A such that (i) B is countably generated,
that is, there is a sequence {Bn , n ≥ 1} with B = σ({Bn , n ≥ 1}) and
(ii) B is equivalent to A, that is, (∀A ∈ A) (∃B ∈ B) µ(A∆B) = 0 and
(∀B ∈ B) (∃A ∈ A) µ(A∆B) = 0.
Conversely if (♠) holds, then show Lp is separable.
Show that both statements above hold even if µ is sigma-finite.
Show that Lp (Rn ) is separable for any sigma-finite measure on Rn .
Give examples of non-sigmafinite (Ω, A, µ) such that
(i) Lp is separable but (♠) fails.
(ii) Lp is non-separable but (♠) holds.
Show L∞ is separable iff there are finitely many ‘distinct’ µ-atoms
whose union is Ω. Recall that atom for µ means a set A ∈ A such
that (i) µ(A) > 0 and (ii) B ∈ A, B ⊂ A implies either µ(B) = 0 or
µ(A \ B) = 0. Two atoms A, B are same if µ(A∆B) = 0.
This is same as saying that L∞ is finite dimensional.
5. Let f be a C ∞ function on R. You know (∃ n)(∀ x)f (n) (x) = 0, then
f is a polynomial. Show that if (∀ x)(∃ n)f (n) (x) = 0. then f is a
polynomial.
Let f be a continuous function on [0, 1]. Let f1 be any function with
f10 = f . In general, fn be any function with fn0 = fn−1 .
Show: if (∃n) (∀x)fn (x) = 0 then f ≡ 0.
Show: if (∀x)(∃n)fn (x) = 0 then f ≡ 0.
6. f ∈ C[0, 1] is said to be everywhere oscillating if it is not monotone on
any (non-degenerate) interval. Can you imagine graph of such a function? Show: Complement of the set of everywhere oscillating functions,
is countable union of small sets.
7. (X, d) be a complete metric space and F be a collection of real continuous functions on X. Assume: (∀ x ∈ X), {f (x) : f ∈ F } is a bounded
subset of R. Show: (∃ U 6= ∅ open) {f (x) : x ∈ U ; f ∈ F } is bounded.
==============
2
CMI/BVR
Graduate Analysis 2 HA 2
10-01-2024
Worry is a kind of thought and memory evolved to give life direction and
protect us from danger. Without its nagging whispers, we would be prone
to a reckless Panglossian life style marked by drug abuse, unemployment,
bankruptcy, · · · . A modest level of worry is usually best. Too much worry
strands us in an agitated state of despair, anxiety and paranoia; too little
leaves us without motivation and direction. Worry contributes to life’s todo list, but its relentless prompts are unpleasant and we work to diminish
them by crossing items off the list.
The bottom line? Stop worrying
about worry. It is good for you.
Robert Provine
8. Set of first category is small (topologically) and set of measure zero is
small (measure theoretically). Exhibit a set A ⊂ R which is small in
Lebesgue sense while Ac is small topologically.
9. You know there is a continuous function x on R whose set of discontinuity points Dx is exactly the set of rationals. (♠) Is there a continuous
x with Dx exactly the set of irrational numbers?
Let x : R 7→ R be any function. For δ > 0 define a function on R to R
by Oδ (t) = sup{|x(s) − x(u)| : t − δ ≤ s, u ≤ t + δ} (O= oscillation)
Show that Oδ ↓ and limit be denoted O(t). Show that x is continuous
at a point t iff O(t) = 0.
Show for each a > 0, the set {t : O(t) < a} is open. Now Answer (♠).
You know that if a sequence (xn ) of continuous functions converges
point-wise to x, then x need not be continuous. (♣) Must it have at
least one point where it is continuous?
If {Jn } is sequence of all intervals with rational end points and An =
x−1 (Jn ), show
Dx ofSdiscontinuity points of x is expressed
S that the set
c
o
c
)
=
(A
−
A
as: Dx =
n
n
n
n (An − An ).
If x is pointwise limit of a sequence of continuous functions and C ⊂ R
is a closed set, Show that x−1 (C) is a Gδ set. Now answer (♣).
Let x be indicator function of the set of rationals. is it pointwise limit
of a sequence (xn ) of continuous functions? Is it pointwise limit of a
sequence of functions (yn ) where each yn is a limit of a sequence of
continuous functions?
10. Show that the set of Polynomials in two variables is dense in the space
C([0, 1]2 → R). Generalize to higher dimensions.
Let D denote the Closed unit disc. Show that Polynomials in z and z
are dense in C(D → R).
3
Let X, Y be compact Hausdorff spaces. Let D1 and D2 be dense subsets
respectively in C(X → R) and C(Y → R). Let A be the set of functions
h(x, y) which are finite sums of functions of the form f (x)g(y) where
f ∈ D1 and g ∈ D2 . Show: A is dense in C((X × Y ) → R).
11. If X is compact metric, show C(X → R) is separable.
Assume X is a compact Hausdorff space. If C(X → R) is separable;
show X is metrizable.
12. Consider the metric space, X = {0, 1}∞ , the infinite product of two
point spaces. A function f : X → R is said to be finite dimensional, if
there is an integer k (depending on f ) such that whenever two points
x, y ∈ X agree in the first k coordinates, then f (x) = f (y). Show that
any such function is continuous. Show that the set of finite dimensional
functions is dense in C(X → R).
13. For x ∈ C[0, 1], let T x be the function T x(t) =
Rt
x(s)ds; 0 ≤ t ≤ 1.
0
Show that the set {T x : kxk ≤ 31} ⊂ C[0, 1] has compact closure.
14. Let K be a continuous function on the closed unit square. For f ∈
R1
C[0, 1] define (Kf )(x) = K(x, y)f (y)dy. Show Kf ∈ C[0, 1].
0
Show that for any bounded subset S ⊂ C[0, 1] the set K(S) ⊂ C[0, 1]
has compact closure.
15. Let (X, d) be a compact metric space and α > 0. Let
K = {f ∈ C(X) : |f (x)| ≤ 39; |f (x) − f (y)| ≤ 85dα (x, y) ∀ x, y ∈ X}
Show K is compact in C(X).
16. Ω ⊂ C, set of complex numbers, is an open connected set and {fn } a
sequence of holomorphic functions on Ω. Assume that for each compact
K ⊂ U , the set {fn (z) : n ≥ 1, z ∈ K} is a bounded set. Show
that there is a subsequence of {fn } which converges to a holomorphic
function, uniformly on compact sets.
==============
4
CMI/BVR
Grduate Analysis 2 HA 3
19-01-2024
A University is a seat of learning, not a centre of worship. It believes in the
pursuit of knowledge and not in the establishment of a cult. As university
men it is our privilege and honour to seek for truth and in this pursuit we
should not be deterred by the fear of what we might find.
S Radhakrishnan
(Times have changed, be careful. if you follow this faithfully, may end up
in jail.)
17. Let X be compact Hausdorff and C(X) is the ring of real continuous
functions on X. Let J be a closed ideal in C(X). If f ∈ C(X) and f
vanishes on the zero set of J, then show f ∈ J. zero set of J means
{x ∈ X : g(x) = 0 ∀g ∈ J}.
[ Given f choose a small neighbourhood of its zero set and a function
g ∈ J strictly positive outside that neighbourhood. f · ng/(1 + ng) ∈ J
and for large n, it is close to f .]
18. C(X,Y) both X, Y compact metric. Formulate and prove Arzela-Ascoli.
19. Many differential equations occurring in practice are second order:
ϕ00 (x) = f (x, ϕ(x), ϕ0 (x)). Formulate Peano/Picard: f : U ⊂ R3 → R2 .
Given (t0 , x0 , y0 ) ∈ U , Looking for a > 0 and h = (ϕ, ψ) : (t0 − a, t0 +
a) → R2 such that h0 (t) = (ψ(t), f (t, ϕ(t), ψ(t)).
Confidence building: try higher order equation.
20. V is real linear space and K ⊂ V be a convex set. We say x is an inner
point of K if there is a small segment from x contained in K in each
direction: (∀ y ∈ V ) (∃ > 0) (∀ t) (|t| < ⇒ x + ty ∈ K).
Assume zero is ‘inner point’ of K. Define ‘Minkowski function’ or
‘gauge function’ of K by p(x) = inf{a > 0 : (x/a) ∈ K} for x ∈ V .
Show p is subadditive and positive homogeneous.
Show: x ∈ K ⇒ p(x) ≤ 1.
Show p(x) < 1 ↔ x is an inner point of K.
Take V = R2 and calculate p when K = {(x, y) : x2 + y 2 ≤ 1};
K = {x2 +y 2 < 1}; K = {x > −1, y > −1}; K = {(x/4)2 +(y/3)2 ≤ 1}.
K nonempty convex, all points of K are inner points. Given y 6∈ K
show there is a linear functional L and real number c such that L(x) < c
for all x ∈ K where as L(y) = c.
5
K convex; with at least one inner point. Suppose y 6∈ K. Show there
is a non-zero linear functional L such that L(x) ≤ L(y) for all x ∈ K
K1 , K2 disjoint non-empty convex sets with K1 having an inner point.
Show that there is a linear functional L and a number c such that
L(x) ≤ c ≤ L(y) for all x ∈ K1 and y ∈ K2 . (use K1 − K2 , the vector
difference set).
For a linear functional L and a number c, the set {x : L(x) = c} is
called a Hyperplane. The set {x : L(x) < c} is called an open halfspace and {x : L(x) ≤ c} is called a closed half space. If there is a norm
on V and L is continuous these are indeed open and closed sets. The
above result is called ‘separation theorem’ for convex sets. Picturize in
R2 and R3 .
21. Let (X, k · k) be a Banach space. If F is a finite subset of X, then show
that convex hull of F is compact.
If C is a compact set, then show that the closed convex hull of C is
compact and convex. Closed convex hull is defined as the closure of
the convex hull of K. ( nets will help.)
22. Review completeness of: C(X) for compact Hausdorff;
lp and Lp (Ω, A, µ) for 1 ≤ p ≤ ∞;
c0 = space of sequences converging to zero, sup norm.
C0 (X) space of continuous functions vanishing at infinity for a LCH
space X.
23. Let (X, k · k) be a normed linear space. Show P
that it is complete iff the
following holds: (xn ) is a sequence in X and
kxn k < ∞ implies the
∞
P
series
xn converges in X.
1
24. Review Zorn’s Lemma/Axiom of choice.
A collection F of subsets of N = {1, 2, · · · } is called a filter if (i) ∅ 6∈ F
and (ii) A, B ∈ F ⇒ A∩B ∈ F and (iii) A ∈ F, A ⊂ B ⊂ N ⇒ B ∈ F.
An ultra filter is a filter not strictly contained in any larger filter. Show
there are ultrafilters.
Show that there is a finitely additive probability µ defined on all subsets
of N such that µ{n} = 0 for all n.
Show that you can define a ‘limit’ L for every bounded sequence such
that (i) L is linear and (ii) L is positive: if xn ≥ 0 for each n, then
L((xn )) ≥ 0 and (iii) L agrees with usual limit when the later exists.
6
CMI/BVR
Graduate Analysis 2 HA 4
26-01-2024
In soils of sand, the more you delve the more rush it springs ♦
So too in learning, the deeper you go the more bounty it brings.
Of the many slips on the slope of life’s slippery slips ♦
The worst is the careless word that passes through your lips.
Tirukkural
25. Complete the details of calculating dual of Rn with lp norms; in particular identify the dual norm. Here 1 ≤ p ≤ ∞. I am not sure if you
did this earlier.
26. Complete the details of completion of normed linear space. First recall
details of completion of metric spaces. The only extra thing you then
need to do is to make sure that vector operations are well defined.
27. Let M ⊂ X be a closed subspace. Let x0 6∈ M . Show d(x0 , M ) > 0.
Show there is L ∈ X ∗ such that kLk = 1 and L(x0 ) = d(x0 , M ) and L
is zero on M .
28. For any subset S ⊂ X, Banach space define
S ⊥ = {x∗ ∈ X ∗ : x∗ (s) = 0 ∀s ∈ S}.
Show S ⊥ = (S)⊥ = (span S)⊥ = span S
⊥
.
29. Show that the unit ball in any normed linear space is convex.
Let 1 < p < ∞ show that the unit ball in Lp [0, 1] is strictly convex:
for any kf kp = kgkp = 1 with f 6= g we have k(f + g)/2k < 1. This
pictorially means that the surface of the unit ball contains no straight
line.
Show this is false for p = 1 and p = ∞ and also for C[0, 1].
Is [0, 1], Lebesgue measure important in the above?
30. Show that every proper closed subspace of a Banach space X is intersection of closed half spaces containing it.
31. Consider X = L2 [−1, 1]. For a ∈ R, let Ka be the set of continuous
functions f on [−1, 1] with f (0) = a. Thus Ka ⊂ X (??).
Show Ka is convex and dense in X. Show that for different a, b these
are disjoint convex sets. Do you think you can separate by hyperplane?
7
32. Let V be the vector space of all real continuous functions on (0, 1).
For f ∈ V and > 0, let N (f, ) = {g ∈ V : |f (x) − g(x)| < ∀x}.
Consider the topology on V generated by these sets. Show that in this
topology addition is continuous, but scalar multiplication is not.
33. Let X be a finite dimensional normed linear space with
P norm x 7→
Pp(x).
Let e1 , . . . , en be a basis. Define q on X by q(x) = |xi | if x = xi ei .
Show q is a norm on X.
Let M = max p(ei ). Show M > 0 and p(x) ≤ M q(x) for all x.
(•).
Plan: to show m > 0 such that p(x) ≥ mq(x) for all x.
(••)
Argue this is same as showing: p(x) ≥ m for all x with q(x) = 1.
n
n
P
P
If false, show, for each k ≥ 1, there is xk =
xki ei with
|xki | = 1
i=1
i=1
and p(xk ) < 1/k. Fix one such xk for each k.
Argue that there is some i so that xki 6→ 0.
Now carefully choosing a subsequence (xkm = ym ) get y 6= 0 so that
p(ym − y) → 0 contradicting p(ym ) → 0.
Conclude (••).
Show that if you have two norms p1 and p2 on a finite dimensional
space then they are equivalent: there are m, M > 0 so that mp(x) ≤
q(x) ≤ M p(x) for all x.
Show: on a finite dimensional space, all norms give the same topology.
Show that every linear functional on a finite dimensional space is continuous.
===================
Quiz: Wednesday February 7 as discussed in class.
======================
8
CMI/BVR
Graduate Analysis 2 HA 5
09-02-2024
With automatic spellcheck, spelling has deteriorated. With smart phone
at fingertips multidigit multiplication in head disappeared. I can not say
whether GPS and navigation software have made my mental maps any
more or less accurate. These are some of the skills partially outsourced
from many minds, thanks to technological advances. And it will not
be long before important aspects of our social and intellectual life follow suit. Forfeiting these skills to gadgets does not worry me so much.
What does worry me is the illusion of knowledge and understanding that
can result from having information so readily and effortlessly accessible.
· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · It follows that smarter and more efficient
information retrieval via machines could foster dumber and less effective
information processing in human minds.
Tania Lombrozo
34. Show that for 1 ≤ p < ∞; we have lp∗ = lq for complex spaces. Also we
have L∗p = Lq for complex spaces over a sigma-finite measure. Remember you have to identify the norms also.
35. Consider Two point set with measure of one point one, other point
measure infinity. Discuss The L1 and L∞ spaces. is the measure space
sigma finite?
Consider [0, 1], sigmafield all sets, measure counting measure: number
of points for finite sets and infinity for others. is this sigma finite?
Discuss L1 and L∞ spaces.
36. When does equality hold in Holder?. Consider 1 ≤ p ≤ ∞ and (complex)
Lp space Rover sigmafinite measure µ. Holder says:
R
| f g|dµ ≤
|f ||g|dµ ≤ kf kp kgkq
(F)
Show that first inequality in (F) is equality iff there is a real number
θ such that f (x)g(x) = eiθ |f (x)||g(x)| for a.e. x
If f ≡ 0 a.e., then second inequality in (F) is an equality. So assume
that f is not the zero function. Show that the second inequality in (F)
is an equality iff there is a real number λ such that:
|g(x) = λ|f (x)|p−1 for a.e. x. in case 1 < p < ∞
|g(x)| ≤ λ and |g(x)| = λ when f (x) 6= 0 in case p = 1
|f (x)| ≤ λ and |f (x)| = λ when g(x) 6= 0 in case p = ∞.
37. If x ∈ la , y ∈ lb and zP∈ lc where 1 < a, b, c < ∞ and (1/a) + (1/b) +
(1/c) = 1 then show
|xn yn zn | ≤ kxka kykb kzkc .
9
38. Let X be a Banach space with dual X ∗ . If X ∗ is separable, then show
that X is separable. (Take countable dense set in X ∗ and pick countable
D ⊂ X on which these elements of X ∗ attain norm).
39. Let X be a Banach space, X ∗ its dual with its norm and X ∗∗ be dual
of X ∗ . Show that each element x ∈ X can be ‘canonically’ identified
with an element of X ∗∗ . Show that the identification is linear isometry
from X into X ∗ .
40. Consider CR (X) (real continuous functions) where X is a compact
Hausdorff space. Let L be a linear functional on C(X) such that
L(1) = kLk. Show that L is positive linear functional.
41. Let c0 be the space of all real sequences converging to zero with sup
norm. Show that c∗0 is the real l1 .
42. Show that closed unit ball of L1 [0, 1] (Lebesgue measure) has no extreme points. Show that every point on the surface of closed unit ball
in Lp [0, 1] is an extreme point of the closed unit ball for 0 < p < 1.
Extreme point of a convex set C is a point v such that if v = λv1 +
(1 − λ)v2 with 0 < λ < 1 and v1 , v2 ∈ C then v1 = v2 . Thus v can not
be written as a non-trivial convex combination of elemenst in C.
43. Show
that L2 [0, 1] ⊂ L1 [0, 1] (Lebesgue measure). Show that {f ∈ L2 :
R
2
|f | ≤ n} is closed in L1 but has empty interior.
Deduce that L2 is a first category set in L1 .
R
Put gn = n on [0, 1/n3 ] and zero outside. Show that f gn → 0 for
every f ∈ L2 but there are f ∈ L1 for which this is not true.
Show that the inclusion map from L2 to L1 is continuous but not onto.
44. Suppose Bn with norm k · kn be a Banach space for each n ≥ 1. Fix
1 < p < ∞. Consider
X
B = {x = (xn : n ≥ 1) : (∀n) xn ∈ Bn ;
kxn kpn < ∞}
Is it a linear space? Is the following a norm?
X
kxk = (
kxn kp )1/p
Is the space complete?
Rutuja pointed out an error in ex 13 which I corrected. The set is not
compact, but has compact closure as in ex 14.
10
CMI/BVR
Graduate Analysis 2 HA 6
16-02-2024
I am worried about stupid, it is all around us. Elected representatives can
not seem to make a reasonable argument. In customer service, the person
you are talking to is reading from a script, can not deviate because they
do not know what they are talking about. News has become a mouth
piece for views that can be parroted by their listeners. Challenging beliefs
is not part of the news anymore. I worry we stopped asking ‘why’. We say
prescription drug works miracles, but we fail to ask what we really know
about what else it does. We glorify stupid on TV shows, showing what
dumb things people do, so that we can all laugh at them. I worry behind
this glorification of stupidity and the refusal to think hard about real issues
are big corporations who make a great deal of money on this. I am worried
that people can not think, can not reason from evidence, and do not even
know what would constitute evidence. People do not know how to ask the
right questions. · · · · · · .
Roger Schank
45. µ is a signed measure. Show that µ+ is the smallest measure above µ:
That is if ν is a measure and ν(A) ≥ µ(A) for all A then ν(A) ≥ µ+ (A).
Show that µ− is the smallest measure above −µ: That is if ν is a
measure and ν(A) ≥ −µ(A) for all A then ν(A) ≥ µ− (A).
Thus, denoting the zero measure by 0, you can say µ+ = µ ∨ 0 and
µ− = (−µ)∨0. Remember f + = f ∨0 and f − = (−f )∨0. For functions
these are point-wise. Interestingly for µ, these are not set-wise. Think
about it.
46. Show that total variation kµk on the collection M of finite signed measures on (Ω, M) is a norm and M is complete under this norm.
If Ω is lch space and M is the collection of finite regular Borel signed
measures then M is also a Banach space.
47. RConsider C[0, 1] (real) and µ be a finite measure on [0, 1] (?). If
xn dµ(x) = 0 for integers n ≥ 1, show that µ is the zero measure.
48. For any finite signed
P measure µ, define |µ| the total variation measure
by |µ|(A) = sup{ |µ(Ai )|} where the sup is over all measurable finite partitions of A. show that this is a finite positive measure. No
calculations needed.
49. Consider Cc (R) and L is the Riemann integral. What is µ?
Consider L(f ) = f (29) − f (33). What is µ?
11
50. X is lch space. Show that Cc (X) is dense in C0 (X) (real). Show that
any positive linear functional on C0 (X) is bounded (You may not have
access to the constant function one). Show that any bounded linear
functional is difference of two positive linear functionals.
51. Consider Rd × R where Rd is reals with discrete topology and R is real
with usual topology. Show that the space is locally compact. How do
compact sets look like? How do Cc functions look like?
R Show that if
f ∈ Cc then for P
allRbut finitely many x, we must have f (x, y)dy = 0.
Define L(f ) =
f (x, y)dy. Discuss the measure µ for L. Is your
x
measure regular?
52. Consider RL2 [0, 1]. Fix K(x, y) in L2 ([0, 1] × [0, 1]). You know that
T f (x) = K(x, y)f (y)dy is a bounded linear map
Pof L2 to itself. This
appears similar to Matrix transform: Av(i) =
a(i, j)v(j) Replace
(i, j) by (x, y); v by f and sum by integral. With your expertise on
matrices,
exhibit a function L in L2 ([0, 1] × [0, 1]) and show T 2 f (x) =
R
L(x, y)f (y)dy. Here T 2 f is T (T f ).
Generalize in two directions: (i) Instead of T 2 , try T1 T2 ; (ii) instead of
stopping at T 2 , try T 3 , T 4 .
53. Suppose v1 , · · · , vn are linearly independent vectors in a (real) Banach
space P
X. Let z ∈ X show that there are numbers a1 , · · · , an such that
kz − ( ai vi )k is minimized.
Consider R2 with l∞ norm. Describe all vectors on x-axis which are
closest to the point (4, 5).
54. Consider l2 . Show that the following set is compact.
1
S = x = (xn ) : |xn | ≤
n
12
CMI/BVR
Graduate Analysis 2 HA 7
21-02-2024
Within a few decades, Big Data algorithms, informed by a stream of biometric data, could monitor our health 24/7. People will enjoy the best
health-care in history, — but for precisely this reason they will probably
be sick all the time. There is always something wrong somewhere in the
body. There is always something that can be improved. In the past, you
felt perfectly healthy as long as you did not sense pain and you did not
manifest an apparent disability by, say limping. But by 2050, thanks to
biometric sensors and Big Data algorithms, diseases may be diagnosed and
treated long before they lead to pain or disability. As a result you will
always find yourself suffering from some “medical condition” following this
or that algorithmic recommendation.
Yuval Noah Harari.
55. Suppose (an ) is a sequence of real numbers.
(i) For every real
P sequence (xn ) which converges to zero, you are told
that the series
an xn converges. Show
P that (an ) ∈ l1 .
(ii) Whenever (xn ) ∈ l1 , the series Pan xn converges. Show (an ) ∈ l∞ .
(iii) Whenever (xn ) ∈ lp the series
an xn converges. Show (an ) ∈ lq .
Here 1 < p < ∞ and q is conjugate exponent.
(iv) Suppose that g is a real measurable
function on [0, 1]. Assume that
R
for each f ∈ Lp [0, 1] the integral f gdx is finite. Show that g ∈ Lq .
(1 < p < ∞).
56. X is a real Banach space and T : B × B → R be a bilinear map.
Suppose that for each x ∈ X, the map y 7→ T (x, y) is continuous and
for each y, the map x 7→ T (x, y) is continuous. Show that the map
(x, y) 7→ T (x, y) is continuous.
57. X and Y are Banach spaces and T is a linear map on X onto Y . You
know that if there is c > 0 such that kT xk ≤ ckxk then T is continuous.
Prove the following: If there is c > 0 such that kT xk ≥ ckxk then T is
continuous.
58. Let 0 < a ≤ 1. The space Lip(a) is the collection of complex functions
f on the interval [23, 89] such that
|f (s) − f (t)|
< ∞.
|s − t|a
s6=t
Mf = sup
13
Define kf k1 = |f (23)| + Mf and kf k2 = sup |f (x)| + Mf . Show that
23≤x≤89
these are norms and the space is a Banach space.
59. Consider a finite measure space (X, A, µ). For f ∈ L∞ (µ) define the
linear map Mf on L2 (µ) by Mf (g) = f g. Calculate kMf k. When is
this map ONTO L2 ?
60. Show that for 1 ≤ p ≤ r then show that lp ⊂ lr
Show that for 1 ≤ p ≤ r then show that Lr [a, b] ⊂ Lp [a, b] for any finite
interval.
For p 6= r, show that there is no inclusion between Lp (0, ∞) and
Lr (0, ∞).
Let 0 < a ≤ b < ∞. Consider the function
f (x) =
1
xa + xb
For which p is this in Lp (0, ∞)
Are the following subspaces closed in l2 ?
M = {x = (xn , n ≥ 1) ∈ l2 :
∞
X
1
n
n=1
xn = 0}
∞
X
1
√ xn = 0}
N = {x = (xn , n ≥ 1) ∈ l2 :
n
n=1
Is the following subspace closed in L2 [0, 1]?
Z 1
x(t)
M = {x = x(t) ∈ L2 [0, 1] :
dt = 0}
t
0
is the following subspace closed in L2 [1, ∞)?
Z ∞
x(t)
M = {x = x(t) ∈ L2 [1, ∞] :
dt = 0}
t
1
61. X and Y are two normed linear spaces and x ∈ X and y ∈ Y . Show
that there is a bounded linear map T : X → Y such that T x = y.
14
CMI/BVR
Graduate Analysis 2 HA 8
08-03-2024
Imagine hundred years from now there is a discussion of Professor Seshadri.
Would you like people to (i) be inspired by his life and work/understand his
theorems · · · OR (ii) fight over whether his birth place was Kanchipuram
or Thanjavur/whether he ate onions · · · .
62. Consider C(X) complex continuous functions on compact metric space
X. Show that its dual isPM space of complex measures with total variation norm. kµk = sup |µ(Ai )| where sup is over all finite partitions
(Ai ) of X.
63. Complete the details: If X is separable Banach space, then norm-closed
unit ball of X ∗ is compact metric space under weak* topology.
64. X is a Banach space and z ∈ X and A ⊂ X. Show that z belongs to
the closed linear span of A iff the following happens:
l ∈ X ∗ , l(y) = 0 ∀ y ∈ A
⇒
l(z) = 0
65. Let T : Cm → Cn given by a matrix in the usual bases. Calculate its
adjoint.
66. Consider lp (real) where (1 ≤ p ≤ ∞). define
T x = (0, x1 , x2 , x3 , · · · );
Sx = (x2 , x3 , x4 , · · · );
x = (x1 , x2 , x3 , · · · )
Show they are bounded operators, calculate their norms, adjoints.
Find if the sequence (T n ) (S n ) are converging in uniform/strong/weak
topologies.
Same question for Lp (R) and T f (x) = f (x + 1) and Sf (x) = f (x − 1).
You have to first show that these are well defined.
67. Consider C[0, 1] (real). Define
Z x
T f (x) =
f (z)dz;
0
Z x
Sf =
z 10 f (z)dz
0
Calculate their norms, Discuss convergence of (T n ), Find T ∗ , S ∗ .
Consider Tn f (x) = xn f (x). Discuss convergence of {Tn }.
68. Show that C[0, 1] is not reflexive. Show that l1 is not reflexive.
X is reflexive Banach space and Y is a closed subspace of X. Show
that Y is a reflexive Banach space.
15
69. For this and next exercise: X is Banach space with dual X ∗ .
For Y , a closed subspace of Banach space X. Y ⊥ , called annihilator of
Y is defined as {l ∈ X ∗ : l(Y ) = 0}. Here l(Y ) = 0 means l(y) = 0 for
all y ∈ Y . Similarly, (but not identically) For Z, a closed subspace of
X ∗ ; Z ⊥ , called annihilator of Z (in X) is defined as {x ∈ X : Z(x) = 0}.
Here Z(x) = 0 means l(x) = 0 for all l ∈ Z.
Show Y ⊥ is a closed subspace of X ∗ .
S and T , are closed subspaces. Show DeMorgan’sLaws:
S ∩ T = (S ⊥ + T ⊥ )⊥ and S ⊥ ∩ T ⊥ = (S + T )⊥ .
Show (S ∩ T )⊥ ⊃ S ⊥ + T ⊥ and (S ⊥ ∩ T ⊥ )⊥ = S + T
70. X is a Banach space and Y a closed subspace.
Show for z ∈ X:
inf{kz − yk : y ∈ Y }
max{|l(z)| : klk ≤ 1, l(Y ) = 0}
=
Show for l ∈ X ∗ :
sup{|l(y)| : y ∈ Y, kyk = 1}
=
min{kl − mk : m ∈ Y ⊥ }
71. Let X be a Banach space (dimension at least 3) and consider L(X, X),
bounded linear operators on X into X. Show that this is a noncommutative ring and kABk ≤ kAkkBk. Show that if kAk = q < 1
Then I − A has an inverse. Find it.
72. Let X be a Banach space. A basis is a sequence (en : n ≥
P1) such that
every element x ∈ X has a unique representation: x =
an en where
an are scalars. Show lp spaces have bases.
73. A sequence (v n ) in lp converges weakly to v iff the sequence is norm
bounded and coordinate-wise they converge.
A sequence fn in Lp [0, 1] (Lebesgue) converges weakly to f iff it is norm
Rt
Rt
bounded and for each t, 0 fn → 0 f .
74. Banach-Mazur Theorem: (real spaces) C[0, 1] is universal for separable
Banach spaces: Every separable Banach space is isometrically isomorphic to a subspace of C[0, 1].
Idea: First try C(B1∗ ) where B1∗ is closed unit ball of dual with weak*
topology; see if you can relate B1∗ to cantor set and Cantor set to [0, 1].
Ojas and Shubham pointed two typos, corrected.
16
CMI/BVR
Graduate Analysis 2 HA 9
15-03-2024
Numbers in India are politicised as never before. The consumer expenditure survey 2017-18 was scrapped because it was of poor quality, we do
not know what was wrong. The periodic Labour force survey was not even
released until after 2019 Lok Sabha elections. Look at the attack on the
head of the agency that provides guidance for the National Family Health
Survey because the survey came out with results that were not to the liking
of the government. Look at the Suppression of many audit reports like the
Clean Ganga Mission. Look at the selective publication of partial reports
not to provide all the usual details; be it Economic survey which came out
more like a propaganda sheet or even the consumption report which does
not give us sufficient details. We had one of the best statistical systems in
the world ..., for this to be Destroyed in a very blatantly politicised way is
a serious problem.
(Jayati Ghosh, Hindu 15-03-2024)
75. Unit ball in every Hilbert space is strictly convex. Show.
Show: in every infinite dimensional Hilbert space there are curves with
the property that every two non-overlapping chords of it are orthogonal.
Curve is a continuous f : [0, 1] → H. A chord corresponding to 0 ≤
u < v ≤ 1 is the vector f (v) − f (u). (Take L2 [0, 1] and do).
Does there exist a Hilbert space whose Hamel basis has dimension ℵ0 ?
There exists a total set in a Hilbert space such that if you remove any
one element it is still a total set but if you remove any two elements it
is no longer total. Show.
A Hilbert space is locally compact iff it is finite dimensional. Show
A Hilbert space is separable iff dimension is ≤ ℵ0 . Show
Every subspace is weakly closed (remember, a strongly closed set need
not be weakly closed in general). Show.
Weak closure of surface of unit ball is all of closed unit ball. Show
Consider a separable Hilbert space. weak topology on bounded sets is
metrizable. weak topology on H is not.
76. Let Mn be the
P space of n × n complex matrices. Trace of A ∗is defined
as T r(A) =
aii if A = (aij ). Show that hA, Bi = T r(B A) is an
inner product. Show it is a Hilbert space.
17
77. Is there an inner product on Cn such that hv, vi = kvk2∞ . Remember
kvk∞ = max |vj |.
2
78. (a) L2 (R, √12π e−x /2 dx) (Hermite Polynomials)
2
2
Define: Hn (x) = ex /2 Dn (e−x /2 ) for n ≥ 0.
show Hn is n-th degree polynomial. They form ONS.
(b) L2 ([−1, 1], dx) (Legendre Polynomials)
n
define Pn (x) = (−1)
Dn [(1 − x2 )n ] for n ≥ 0.
n!2n
Show that Pn is polynomial of degree n, they form ONS.
(c) L2 ((0, ∞), e−x dx). (Laguerre polynomials)
Define Pn (x) = n!1 ex Dn (e−x xn ) for n ≥ 0.
Show that Pn is polynomial of degree n and they form ONS.
√
(d) L2 ([−1, 1], 1 − x2 dx) (Chebyshev Polynomials)
Consider the expansion
∞
X
1
un (x)tn
=
1 − 2xt + t2
0
Show that un is a polynomial of degree n. This is also ONS but not
easy.
(e) L2 [0, 1] (Haar basis)
Define ϕ(t) = 1 for 0 ≤ t ≤ 1/2 and ϕ(t) = −1 for 1/2 ≤ t < 1 and
ϕ(t) = 0 for t ∈ R \ [0, 1). For n ≥ 0 and 0 ≤ k ≤ 2n − 1 define on [0, 1]
ϕn,k = 2n/2 ϕ(2n t − k)
Show that the family {ϕn,k } along with constant function e0 ≡ 1 is
CONS.
ONS in (a,b,c, d) are also CONS; you may not be able to show.
79. (VItali Theorem) Let (fn ) be an ONS in L2 [a, b]. Show that it is complete iff
∞ Z x
2
X
fn (t)dt = x − a;
∀x ∈ [a, b].
1
a
80. Let (vk ) be a sequence of elements in a Banach space X and (ak ) a
sequence of scalars. Show that the following are equivalent:
(i) there is a linear functional L on X with L(v
Pk ) = ak and kLk
P =M
(ii) for any n ≥ 1 and any scalars c1 , . . . , cn ; | n1 cj aj | ≤ M k n1 cj vj k.
18
CMI/BVR
Graduate Analysis 2 HA 10
22-03-2024
What was never understood was that rights are acquired and that powers are granted. This misconception has displaced ’we the people’, the
grantors, to the status of ’we the other people’, subjects rather than citizens in our own democracy.
Kannabiran
81. A be an (bounded linear) operator from a Hilbert space H into a Hilbert
space K. Define A∗ as we did in Banach spaces. If K = H then this is
what we defned in class.
Show Ker(A) = range(A∗ )⊥ and Ker(A)⊥ = range(A∗ ). Show A is
injective iff range(A∗ ) is dense in H.
82. Let A ∈ B(H). Show that the following are equivalent.
(i) kAxk = kxk for every x ∈ H. (called isometry)
(ii) A∗ A = I .
(iii) hAx, Ayi = hx, yi for every x, y ∈ H.
Give an example to show that A∗ A = I does not imply AA∗ = I. Can
this happen in finite dimensional case?
83. Let A ∈ B(H). Show that the following are equivalent.
(i) AA∗ = A∗ A (Then A is called normal).
(ii) kAxk = kA∗ xk for every x ∈ H.
If A satisfies any of the above then show ker(A) = ker(A∗ ).
84. Let A ∈ B(H). Show that the following are equivalent.
(i) A is isometry and surjective.(called Unitary )
(ii). A∗ A = AA∗ = I
(iii) A is isometry and normal.
85. For the multiplicative operator M f = ϕf on L2 of a σ-finite measure
space, show kM k = kϕk∞ . Show that M is normal. when is M = M ∗
(called self adjoint)?
86. Consider K = H ⊕ H. Suppose A ∈ B(H). Define on K an operator B
by
0
iA
B=
−iA∗ 0
Prove kAk = kBk and B ∗ = B.
87. Let
∞
P
an z n be a power series with radius of convergence R, 0 < R < ∞.
0
If A ∈ B(H) and kAk < R show that there is an operator T ∈ B(H)
19
P
P
such that for any x, y ∈ H; hT x, yi =
an hAn x, yi. If f (z) = an z n ,
then we can refer P
to this T as f (A).
Show that kT − n0 ak Ak k → 0. Show that for an operator B, if
AB = BA then T B = BT .
88. (polarization identity) n ≥ 3 and gcd(k, n) = 1 and ω = e2πik/n . Show
n
1X j
ω kx + ω j yk2
hx, yi =
n j=1
integral version. Show
1
hx, yi =
2π
Z 2π
eiθ kx + eiθ k2 dθ
0
89. The gram determinant of n vectors is defined by
hx1 , x1 i hx1 , x2 i
hx2 , x1 i hx2 , x2 i
G(x1 , x2 , . . . , xn ) = det
..
..
.
.
hxn , x1 i hxn , x2 i
···
···
...
hx1 , xn i
hx2 , xn i
..
.
···
hxn , xn i
Show G ≥ 0. Show (xi ) are linearly dependent iff G = 0.
If M is span of (xi ) and x ∈ H, show
d(x, M ) = inf kx − yk =
y∈M
20
G(x, x1 , x2 , . . . , xn )
G(x1 , x2 , . . . , xn )
1/2
CMI/BVR
Graduate Analysis 2 HA 11
28-03-2024
Furious with the humiliating 142 rank in the World Press Freedom Index, the Union cabinet secretary set up a monitoring committee. There
were 11 bureaucrats and Government-controlled-institution researchers in
a committee of 13 and just 2 journalists. The draft report reflected nothing
of the serious issues raised in the meetings. So I submitted an independent
or dissenting note for inclusion in it. At once, the report, the committee,
everything – vanished. RTI enquiries failed to unearth the report – on
freedom of press. The original exercise was not even investigative journalism – it was investigating journalism, as functioned in India. And it
disappeared at the drop of a dissent note.
P. Sainath
90. (i) Consider L2 [0, 1] and multiplication operator: f (t) 7→ ϕ(t)f (t).
Show this is compact iff f = 0 a.e.
(ii) Consider multiplication operator on l2 : (xn , n ≥ 1) 7→ (an xn : n ≥
1). Show this is compact iff lim an = 0
R
(iii) Show that the Kernel operator Kf (t) = K(t, s)f (s)ds is compact on L2 [0, 1] where K ∈ L2 ([0, 1] × [0, 1]).
(iv) As a special case of above take K(x, y) = (x−1)y for 0 ≤ y ≤ x ≤ 1
and K(x, y) = x(y − 1) for 0 ≤ x ≤ y ≤ 1. Show that the function
u = Kf satisfies u00 = f ; u(0) = u(1) = 0.
(v) Let K(s, t) = ±1 according as t < s or t > s. Show that the eigen
values and eigen vectors of the operator are given by
λn =
2
;
(2n + 1)π
en (t) = ei(2n+1)πt
91. Show that every bounded sequence in a Hilbert space has a weakly
convergent subsequence. Show that an operator is compact iff it transforms weakly convergent sequences into norm convergent sequences.
92. For any (bounded) operator A on H show that the sum
eA = I + A +
A3
A2
+
+ ···
2!
3!
makes sense in uniform norm topology. Show that this operator commutes with A.
If kAk < 1 show that (I − A) is invertible. Show that the set of invertible operators is an open subset of B(H) in the norm topology.
You probably did this in matrix case.
21
93. (i) Suppose that A is a self adjoint operator on a Hilbert space with
A2 compact. Show that A is compact.
(ii) Let (en , n ≥ 1) be CONS in H. Define
Ax =
∞
X
hx, e2n+2 ie2k
n=1
Show A is bounded, A is not compact, A2 is compact.
P
94. Le (an ). be a sequence with
|an | finite. Define linear operator on l2
using the matrix


a1 a2 a3 a4 · · ·
 a2 a3 a4 a5 · · · 


 a3 a4 a5 a6 · · · 


.. .. .. ..
..
. . . .
.
Show that A is bounded operator and is compact.
95. Consider the multiplication operator on L2 [0, 1]: Af (t) = tf (t). Does
it have eigen values? is it self adjoint? Does it have a square root?
96. An operator
PA on a2 separable Hilbert space H is said to be HilbertSchmidt if
kAen k < ∞ for one CONS (en ). In that case it is finite
for any CONS and has the same value. L2 (H) denotes the collection
of all H-S operators.
pP
Show that this space is a(complex) linear space and kAk2 =
kAen k2
is a norm.
Show kA∗ k2 = kAk2 and thus the space is closed under adjoint operation.
Show that kAk ≤ kAk2 , where kAk is the usual operator norm.
For A, B ∈ L2 (H), Show kABk2 ≤ kAk kBk2 and also kABk2 ≤
kAk2 kBk.
In fact show that L2 (H) is a Hilbert space with inner product hhA, Bii =
P
hAen , Ben i for some CONS. Show this sum does not depend on the
CONS used.
P
Show multiplication operator: (xn ) 7→ (an xn ) is H-S iff
|an |2 < ∞.
(may not be easy) Show that Kernel operator on L2 [0, 1] with Kernel
K ∈ L2 ([0, 1] × [0, 1]) is H-S.
22
CMI/BVR
Graduate Analysis 2 HA 12
28-03-2024
I have often pondered over the roles of knowledge or experience, on the
one hand, and imagination or intuition, on the other, in the process of
discovery. I believe that there is a certain fundamental conflict between
the two and knowledge, by advocating caution, tends to inhibit the flight
of imagination. Therefore a certain naiveté, unburdened by conventional
wisdom, can sometimes be a positive asset.
Harish-Chandra
97. (I should be talking about nets, but do not want to burden you. We
only consider sequences, which is however not good. You get the spirit.)
(i) In L2 (R) multiplication by I[−n,+n] converges in SOT (strong operator topology) to Identity I, but not in norm. Show
(ii) On l2 , U n converges to zero in SOT but not in norm. Here
U (x1 , x2 , x3 , · · · ) = (x2 , x3 , x4 , · · · ).
(iii) on l2 , S(x1 , x2 , . . .) = (0, x1 , x2 , · · · ) then S n converges to zero in
WOT, not in SOT.
(iv) Show that both WOT and SOT respect addition and scalar multiplication in B(H).
(v) Adjoint map A 7→ A∗ is continuous in WOT but not in SOT. However among normal operators adjoint is continuous in SOT.
(vi) If An → A in SOT, show kAk ≤ lim sup kAkn .
(vii) Show: An → A SOT iff (A − An )∗ (A − An ) → 0 in WOT.
fn (n ≥ 1) and f are in L∞ of a σ-finite measure space. (viii) Show
fn → f in WOT regarded as multiplication operators in B(L2 ) iff
fn → f as elements of L∗1 , the dual of L1 .
98. Consider H = C2 . Consider the matrices:
1 1 −i
1 1 1
1 0
0 0
;
;
;
;
0 0
0 1
2 1 1
2 i 1
Show that they are all projections and they span all 2 × 2 matrices.
99. Let f ∈ L∞ ⊂ B(L2 ) of a σ-finite measure space. Show that f is a
projection iff f = IC for some (measurable) set C.
100. Suppose A : S ⊂ H → H be inner product preserving: hAx, Ayi =
hx, yi for x, y ∈ S. Show that you can extend A as a linear isometry
with domain closed span of S. Note: I did not say whether S is linear
or A is linear.
23
P
101. (i) If n1 (xi ⊗ yi ) = 0 and yi linearly independent then xi = 0 for all i.
(ii) Show that the map x ⊗ y 7→ y ⊗ x extends to a unitary map to all
of H⊗2 and is unitary.
(iii) If xn → x and yn → y then xn ⊗ yn → x ⊗ y.
(iv) A, B ∈ B(H) then (A ⊗ B)(x ⊗ y) = Ax ⊗ By extends to all of the
tensor product. Denote this by A ⊗ B. Show
(A1 + A2 ) ⊗ B = (A1 ⊗ B) + (A2 ⊗ B);
(λA) ⊗ B = λ(A ⊗ B)
A ⊗ (B1 + B2 ) = (A ⊗ B1 ) + (A ⊗ B2 );
A ⊗ (λB) = λ(A ⊗ B)
I ⊗ I = I;
(A ⊗ B)(C ⊗ D) = (AC) ⊗ (BD)
(A ⊗ B)∗ = A∗ ⊗ B ∗ ;
kA ⊗ Bk = kAkkBk
A ⊗ B is invertible iff both A, B are and then (A ⊗ B)−1 = A−1 ⊗ B −1 .
If A, B compact, then so is A ⊗ B. If A, B are compact hermitian/
compact normal then so is A ⊗ B and
σp (A ⊗ B) = {ab : a ∈ σp (A); b ∈ σp (B)}
If A, B unitary, show that so is A ⊗ B.
(v) If g 7→ Ug and g 7→ Vg are unitary representations of G in H, then
so is g 7→ Ug ⊗ Vg in H ⊗ H.
If g 7→ Ug and g 7→ Vg are unitary representations of G1 and G2 in H1
and H2 respectively, then (g, h) 7→ Ug ⊗ Vh is a unitary representation
of G1 × G2 in H1 ⊗ H2 .
102. Bounded operators on the direct sum H ⊕ H can be regarded as 2 × 2
matrices with entries in B(H). Explain.
24
CMI/BVR
Graduate Analysis 2 HA 13
10-04-2024
Our Statistical system set up by P C Mahalanobis:
No country, developed, underdeveloped or overdeveloped has such a wealth
of information about its people as India in respect of · · · , employment,
unemployment, agricultural and Industrial production.
Edwards Deming of USA;
the experience of India will serve as a guidance and an example worth imitating.
You Poh Seng of China
Present system: reflects Orwellian saying: the authority told you to reject
the evidence of your eyes and ears. It was their final, most essential command.
103. Given A ∈ B(H) show that there is a sequence of finite rank operators
An such that An → A in SOT. Remember H is separable.
104. Let U : H → H be unitary. Define ρ : B(H) → B(H) by: ρ(A) =
U AU −1 . Show
kρ(A)k = kAk; (ρ(A))∗ = ρ(A∗ ); ρ is an algebra isomorphism; ρ(A)
is compact iff A is compact; AB = BA iff ρ(A)ρ(B) = ρ(B)ρ(A); M
is invariant for A iff U M is invariant for ρ(A); M reduces A iff U M
reduces ρ(A).
Two operators A and B are said to be unitarily equivalent if there is a
unitary U such that B = U AU −1 .
105. Show projections P1 , P2 are unitarily equivalent iff Ranges of P1 and P2
have same dimension and kernels of P1 and P2 have same dimension.
106. Af (x) = xf (x) on L2 [0, 1].√ Show that A and A2 are unitarily equivalent. (check f (x) 7→ f (x2 ) 2x is a unitary)
P
107. Suppose
|an | < ∞. Show
n≥0
A=
a0
a1
a2
a3
..
.
a1
a2
a3
a4
..
.
25
a2
a3
a4
a5
..
.
a3
a4
a5
a6
..
.
···
···
···
···
..
.
is a bounded operator on l2 (N ∪ {0}). Sometimes it is profitable to use
(an , n ≥ 0) rather than (a
n , n ≥ 1). For example when you want to
P
take help of power series
an z n .
n≥0
108. Let
P
|an | < ∞.
n≥0
A=
a0
a1
a2
a3
a4
..
.
0 0 0 0
a0 0 0 0
a1 a0 0 0
a2 a1 a0 0
a3 a2 a1 a0
.. .. .. ..
. . . .
···
···
···
···
···
..
.
is a bounded operator on l2 . Show that it commutes with shift: (x0 , x1 , x2 , . . .) 7→
(0, x0 , x1 , x2 , . . .).
109. A Jacobi matrix is a (infinite) matrix of the form (all a, b, c ∈ C)
J=
a0 b 0 0 0 0
c 0 a1 b 1 0 0
0 c 1 a2 b 2 0
0 0 c 2 a3 b 3
.. .. .. .. ..
. . . . .
···
···
···
···
..
.
Show that J is a compact operator on l2 iff an → 0 and bn → 0 and
cn → 0.
110. Show that
A=
1
0
0
0
..
.
1
1
0
0
..
.
0
1
1
0
..
.
0
0
1
1
..
.
0
0
0
1
..
.
···
···
···
···
..
.
0
0
0
1
0
..
.
···
···
···
···
···
..
.
is a bounded operator on l2 . Find kAk.
111. Toeplitz matrix:
A=
0
1
0
0
0
..
.
1
0
1
0
0
..
.
26
0
1
0
1
0
..
.
0
0
1
0
1
..
.
is a bounded operator on l2 and kAk ≤ 2. Taking the vectors xn which
is: one n times followed by zeros ; show that kAk = 2.
112. Consider the Hilbert matrix: A = (aij : i, j ≥ 0) where aij = 2−(i+j+1) .
Show this is a bounded operator on l2 and kAk = 2/3.
(Hint: If u = (1/2, 1/22 , 1/23 , · · · ) then Ax = 2hx, uiu).
∞
P
an z n is a power series with
0
P
radius of convergence at least one, show that f (A) = an An converges
in operator norm. (Here A0 = I). Show cos2 A + sin2 A = I.
113. If A ∈ B(H) with kAk < 1 and f (z) =
114. (un ) is a CONS.
P
(i) If (vn ) are vectors with
kun − vn k2 < 1 show closed span of
(vn , n ≥ 1) is H.
P
(ii) If (vn ) is ONS with
kun − vn k2 < ∞ then show that (vn ) is also
CONS.
115. If An → A in SOT, show kAk ≤ lim sup kAn k. Give example where
equality fails in this.
116. Show {I}0 = B(H) and (B(H))0 = {CI}.
Fix a sigmafinite measure space. Regard L∞ ⊂ B(L2 ), multiplication
operators. Show (L∞ )0 = L∞ .
27
CMI/BVR
Graduate Analysis 2 HA 14
19-04-2024
the most important scientific tool of all is not anything you can buy. It
is you own mind. Your thoughts and ideas are the keys that can unlock
the mysteries. In the search for understanding; ‘questions’ are perhaps the
most powerful force of all.
Jack White
117. Suppose that U is an isometry of H, recall this means kU xk = kxk for
all x. Show this is same as U ∗ U = I. (warning: this does not mean
U U ∗ = I). Show that U x = x iff U ∗ x = x. Use
kU x − xk2 = kU xk2 − hx, U xi − hU x, xi + kxk2
Let M = {x : U x = x} and N is the closed linear span of {x − U x : x ∈
H}. Show H = M ⊕ N. Prove von Neumann mean ergodic theorem:
n−1
1X k
U → PM
n 0
(SOT )
We discussed this for Unitary U in the class.
118. Consider Ω = [0, 1) with Borel sigma field and Lebesgue measure. Define T x = 2x mod (1). Show T is onto, not one-one, but measurable,
and measure preserving: λ(A) = λ(T −1 A). Show that U f (x) = f (T x)
is an isometry of L2 and mean ergodic theorem applies.
Consider same space, but T x = 2x for x < 1/2 and T x = 2(1 − x) for
1/2 ≤ x < 1. This is called tent map (why?). Show this is also onto,
not one-one, measurable and measure preserving. Show U defined as
above is an isometry and von Neumann theorem applies.
119. I gave an example with RZ and infinite product measure. That was out
of syllabus because we did not do infinite measures. However, consider
Ω = {0, 1}Z and finite dimensional cylinder sets and the sigma field
as described in class. For any probability ν on {0, 1} show existence
of infinite product measure. Why is this different from earlier case?
Because this is compact and countable additivity on field is achieved
by just observing finite additivity!
There are other interesting maps but this is not course in ergodic theory.
120. Show that the range of an isometry of a Hilbert space is closed subspace.
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121. You know that evaluation f 7→ f (t) does not make sense in L2 [0, 1].
However one may define on the subspace C[0, 1] and try to extend.
Show that this is not possible by considering:
fn (t) = (t/x)n for 0 ≤ t ≤ x and fn (t) = [(1−t)/(1−x)]n for x ≤ t ≤ 1.
122. Denote symmetrization of the tensor u1 ⊗u2 ⊗· · ·⊗un by u1 ou2 o · · · oun
and the antisymmetrization by u1 ∧ u2 ∧ · · · ∧ un . Let A be the matrix
(hui , vj i)1≤i,j≤n . show
hu1 ou2 o · · · oun , v1 ov2 o · · · ovn i = Permanent(A)/n!
hu1 ∧ u2 ∧ · · · ∧ un , v1 ∧ v2 ∧ · · · ∧ vn i = Determinant(A)/n!
Show that (Do for n = 2 and n = 3 first)
u1 ou2 o · · · oun =
X
1 X
.
.
.
(
i ui )n
1 2
n
2n =±1,∀i
i
Deduce that {un : u ∈ H} spans Hn .
Show that (Do for n = 1, 2 first)
dn
u = n e(tu)|t=0
dt
n
Deduce that {e(u) : u ∈ H} spans Γs (H). Show that the map: H 7→
Γs (H) defined by u 7→ e(u) is a continuous map.
Deduce that if S is a dense subset of H then {e(u) : u ∈ S} spans
Γs (H).
123. Let D ⊂ H1 and ϕ : D → H2 such that hx, yiH1 = hϕ(x), ϕ(y)iH2 .
Show that ϕ extends as an isometry from closed span of D into H2 .
124. If x ∈ H and M is a subspace, then show
d(x, M) = sup{|hx, yi| : y ∈ M⊥ ; kyk = 1}
! ! END ! !
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