FUNCTIONAL ANALYSIS:MA 9201
Short Answer Type Questions (2/3/4 marks)
(1) Show that {x1 , x2 , ..., xn }, where xj (t) = tj , is a linearly independent
set in the space C[a, b].
(2) Show that in an n dimensional vector space X, the representation of any
vector x as linear combination of given basis vectors e1 , e2 , ..., en is unique.
(3) Let {e1 , e2 , ..., en } be a basis for a complex vector space X. Find basis
for X regarded as real vector space. What is the dimension of X in either
case?
(4) If Y andZ are subspaces of a vector space X, show that Y ∩ Z is a
subspace of X but Y ∪ Z need not be one. Give examples.
(5) If M 6= φ is any subset of a vector space X, show that spanM is a
subspace of X.
(6) Show that the set of all real two-rowed square matrices forms a vector
space X. What is the zero vector in X? Determine dimX. Do the symmetric
matrices x ∈ X form a subspace of X? The singular matrices?
(7) Define norm k · k on a vector space X. For x ∈ R3 , show that kxk
represents the length of vector x from the origin.
(8)Show that every normed space (X, k · k) is a metric space, metric d on
X induced by d(x, y) = kx − yk. Is the converse true? Justify your claim.
(9) Prove that norm k · k : X → [0, ∞) is a continuous function.
(10) Let X be a vector space of all ordered pairs x = (ξ1 , ξ2 ), y = (η1 , η2 )...of
real numbers. Show that norms on X are defined by kxk1 = |ξ1 |+|ξ2 |, kxk2 =
p
(ξ12 + ξ22 ), kxk∞ = max{|ξ1 |, |ξ2 |}
(11) Define Banach space. In a normed linear space X, if absolute convergence of a series always implies convergence of that series, show X is a
Banach space .
(12) When do you say that two norms on a vector space are equivalent?
Show that on a finite dimensional vector space any two norms are equivalent.
(13) If two norms k · k1 and k · k2 on a vector space are equivalent, show
that kxn − xk1 → 0 if and only if kxn − xk2 → 0 .
(14) Define linear operator from one vector space to another vector space.If
the product (the composition) of two linear operators exists, show that it is
also linear .
(15) Let T : D(T ) → Y be a linear operator whose inverse exists. If
{x1 , x2 , .., xn } is linearly independent set in D(T ), show that the set {T x1 , T x2 , ..., T xn }
is linearly independent in Y .
(16) Define bounded linear operator.Show that a linear operator on a finite
dimensional vector space is always bounded.
(17) Let X be a normed space of all polynomials defined on J = [0, 1] with
norm kxk = M axt∈J |x(t)|. Define T : X → X by T x(t) = x0 (t). Show that
T is linear but not bounded.
(18) Let X and Y be normed linear spaces. Show that a linear operator
T : X → Y is bounded if and only if T maps bounded sets in X into bounded
sets in Y .
(19) If T 6= 0 is a bounded linear operator , show that for any x ∈ D(T )
such that kxk < 1 we have the strict inequality kT xk < kT k.
(20) Show that the inverse T −1 : R(T ) → X of a bounded linear operator
T : X → Y need not be bounded.
(21) Let T : C[0, 1] → C[0, 1] be defined by
Z
y(t) =
t
x(τ )dτ.
0
Find R(T ) and T −1 : R(T ) → C[0, 1]. Is T −1 linear and bounded?
(22) If Y is a subspace of a vector space X and f is a linear functional on
X such that f (Y ) is not the whole scalar field of X, show that f (y) = 0 for
all y ∈ Y .
(23) Let X and Y be normed spaces. What do you mean by the space
B(X, Y )? Show that B(X, Y ) is a normed space. Define dual space of X.
(24) If X is the space of ordered n − tuples of real numbers and kxk =
maxi |ξi |, where x = (ξ1 , ..., ξn ), what is the corresponding norm on the dual
space X 0 ?
(25) State uniform boundedness theorem. Use it to show that the normed
space X of all polynomials with norm kxk = maxj |αj | (αj are coefficients
of x) is not complete.
(26) Define open mapping from a normed space X into a normed space Y .
Show that a continuous map is not necessarily an open map. Justify with
examples. Also show that an open mapping need not to map closed sets
onto closed sets.
(27) Show that T : R2 → R defined by T (x, y) = x is open. Is the mapping
T : R2 → R2 defined by T (x, y) = (x, 0) an open mapping?
(28) Define closed linear operator from a normed space X into another
normed space Y . Give examples to justify that (i) a closed linear operator
may not be bounded (ii) a bounded linear operator may not be closed. Also
state condition under which a closed linear operator is bounded (Closed
graph theorem).
(29) If the inverse T −1 of a closed linear operator exists, show that T −1 is
a closed linear operator.
(30) Let X and Y be normed spaces. If T1 : X → Y is a closed linear
operator and T2 ∈ B(X, Y ), show that T1 + T2 is a closed linear operator.
(31) Define inner product on a vector space X and norm on X induced
by the inner . product. Show that this norm always satisfies parallelogram
equality:
kx + yk2 + kx − yk2 = 2(kxk2 + kyk2 );
x, y ∈ X.
(32) When do you say that in an inner product space X, two vectors x, y ∈
X are orthogonal (x⊥y). Show that x⊥y ⇒ kx+yk2 = kxk2 +kyk2 . Give an
example to show that converse is not true in general. In what case converse
is also true?
(33) If an inner product space is real, show that the condition kxk = kyk ⇒
hx + y, x − yi = 0. What does this mean geometrically if X = R2 What does
the condition imply if X is a complex vector space?
(34) Verify by direct calculation that for any elements in an inner product
space,
1
1
kz − xk2 + kz − yk2 = kx − yk2 + 2kz − (x + y)k2 .
2
2
Show that this identity can also be obtained from parallelogram equality.
(35) If X is a real inner product space, verify that:
1
hx, yi = (kx + yk2 − kx − yk2 ).
4
(36) If X is a complex inner product space, verify that:
1
<hx, yi = (kx + yk2 − kx − yk2 );
4
1
=hx, yi = (kx + iyk2 − kx − iyk2 ).
4
(37)Let X be an inner product space.Show that for x, y ∈ X,
|hx, y, i| ≤ kxk · kyk.
Also show that equality in above (Schwarz’s inequality) occurs if and only
if the set {x, y} is linearly dependent.
(38) Let X be an inner product space. Prove the triangle inequality :
kx + yk ≤ kxk + kyk
Also show that equality holds if and only if y = 0 or x = cy, where c is real
and ≥ 0.
(39) If X is an inner product space, show that if xn → x and yn → y, then
hxn , yn i → hx, yi. Apply it to show that y⊥xn and xn → x ⇒ x⊥y .
(40) Show that for a sequence {xn } in an inner product space X, the conditions kxn k → kxk and hxn , xi → hx, yi ⇒ xn → x.
(41) Show that in any inner product space X, x⊥y if and only if kx + αyk =
kx − αyk for all scalrs α.
(42) Define orthogonal complement A⊥ of a subset A of a Hilbert space X.
Show that A⊥ is a closed linear subspace of X.
(43) Let Y be a closed subspace of a Hilbert space H. Prove that H =
Y ⊕ Y ⊥.
(44) If Y is a closed subspace of a Hilbert space X, show that Y = Y ⊥⊥ .
(45) Define orthonormal set in an inner product space X. Show that an
orthonormal set is linearly independent.
(46) Let {ek } be an orthonormal sequence in a Hilbert space H. Show that
P
P∞
P∞
if x = ∞
j=1 αj ej , y =
j=1 βj ej , then hx, yi =
j=1 αj βj , the series being
absolutely convergent.
(47) If {ek } is an orthonormal sequence in an inner product space X, and
P
x ∈ X, show that x − y with y given by y = nk=1 αk ek is orthogonal to the
subspace Yn = span{e1 , e2 , ..., en }.
(48) Let {ek } be an orthonormal sequence in an inner product space X.
P
Show that for x, y ∈ X, ∞
k=1 |hx, ek ihy, ek i| ≤ kxkkyk.
(49) Orthonormalize the set {1, x, x2 } on the interval [−1, 1], where hx, yi =
R1
−1 x(t)y(t)dt.
(50) What do you mean by Hilbert basis (orthonormal basis) and Hilbert
dimension of a Hibert space X. Show that in every Hilbert space H 6= {0},
there exists a Hilbert basis or orthonarmal basis.
(51) State and prove Reiz’s Theorem ( Every bounded linear functional
on a Hilbert space H can be represented in terms of the inner product;
f (x) = hx, zi, where z depends on f , is uniquely determined by f and has
the norm kzk = kf k.
(52) If z is any fixed element of an inner product space X, show that
f (x) = hx, zi defines a bounded linear functional f on X of norm kzk. Also
show that if the mapping from X to X 0 given by z 7→ f is surjective, then
X must be a Hilbert space. Here X 0 is dual space of X.
(53) Define sesquilinear form and give one example. State Riesz representation theorem.
(54) Define Hilbert adjoint operator T ∗ of a bounded linear operator T :
H1 → H2 , wher H1 , H2 are Hilbert spaces over the same field. Show that
T ∗ is bounded. What is its norm.
(55) Let X be a complex inner product space and Q : X → X such that
hQx, xi = 0, ∀x ∈ X. Then show that Q = 0. Give an example to show that
result is not true if X is a real inner product space.
(56) Let Tn be a sequence of bounded linear operators on a Hilbert space
such that Tn → T . Show that Tn∗ → T ∗ .
(57) Let T1 and T2 be bounded linear operator on a complex Hilbert space
H into itself. If hT1 x, xi = hT2 x, xi for all x ∈ H, show that T1 = T2 .
(58) Show that for any bounded linear operator T on a Hilbert space H,
the operators T1 = 21 (T + T ∗ ) and T2 =
1
2i (T
− T ∗ ) are self adjoint.
(59) If S andT are normal linear operators such that ST ∗ = T ∗ S and
T S ∗ = S ∗ T , show that their sum S + T and product ST are normal.
(60) Let {Tn } be a sequence of self adjoint operators Tn : H → H on a
Hilbert space H. If Tn → T , then show that T is self adjoint.
(61) Show that in a finite dimensional space, weak convergence implies
strong convergence.
(62) If z is a fixed element of a normed space X and f ∈ X 0 (Dual space of
X), show that the operator T : X → X defined by T x = f (x)z is compact.
(63) IfX is an inner product space, show that T x = hx, yiz with fixed y and
z defines a compact linear operator.
(64) If a normed space X is reflexive, show that it is complete and hence a
Banach space.
(65) Show that every finite dimensional space is reflexive.
Very short answer type questions (1/2 marks)
(1) Which of the following subsets of R3 constitute a subspace of R3 ? [Here
x = (x1 , x2 , x3 )]:
(a) All x with x1 = x2 and x3 = 0 (b) All x with x1 = x2 + 1
with positive x1 , x2 , x3
(c) All x
(d) All x with x1 − x2 + x3 = k (constan).
(2) T : R2 → R2 defined by T (x, y) = (x, 0) is an open mapping (True/Flase).
(3) Let X and Y be normed spaces. Show that a linear operator T : X → Y
is bounded if and only if T maps bounded sets in X into bounded sets in Y .
(4) If T 6= 0 is a bounded linear operator, show that for any x ∈ D(T ) such
that kxk < 1, we have strict inequality kT xk < kT k.
(5) Give an example to show that the inverse T −1 : R(T ) → X of a bounded
linear operator T : X → Y need not be bounded.
(6) Let T : X → Y be a closed linear operator. If T −1 exists, then show
that it is also closed linear operator.
(7) (a) Let x 6= 0, y 6= 0. If x⊥y, show that the set {x, y} is linearly independent. (b) Extend the result to mutually orthogonal non zero vectors,
x1 , x2 , ..., xn .
(8) If in an inner product space, hx, ui = hx, vi for all x, show that u = v.
(9) Which of the following inner product spaces with usual inner product
defined on them is not a Hilbert space:
(a) Rn
(b) Cn
(c) C[a, b]
(d) l2
(10) Give an example of a Banach space which is not a Hilbert space.
(11) Show that the space C[a, b] of all real valued continuous functions defined on [a, b] with norm, kxk = maxt∈[a,b] |x(t)|, is a Banach space but not
a Hilbert space.
(12) Show that an orthonormal set in a normed space is linearly independent.
(13) Let {ek } be an orthonormal sequence in an inner product space X.
P
Show that for any x, y ∈ X, ∞
k=1 |hx, ek ihy, ek i| ≤ kxkkyk.
P
(14) Illustrate with an example that a convergent series hx, ek iek need
not have sum x.
(15) What do you mean by Hilbert basis of a Hilbert space H 6= {0}?
(16) If z is any fixed element of an inner product space X, show that
f (x) = hx, zi defines a bounded linear functional f on X, of norm kzk.
(17) If T : H1 → H2 be a bounded linear operator from Hilbert space H1
to Hilbert space H2 , show that kT ∗ T k = kT T ∗ k = kT k2 .
(18) Show that the product of two self adjoint operators is self adjoint iff
operators commute.
(19) Let T : H → H be a bounded linear operator on a Hilbert space H.
Show that if T is self adjoint, then hT x, xi is real for all x ∈ H.
(20) Let T : H → H be a bounded linear operator on a Hilbert space H.
Show that if H is complex and hT x, xi is real for all x ∈ H, then the operator T is self adjoint.
(21) If U : H → H is unitary operator on a Hilbert space H, show that
kU k = 1 and also show that U −1 is unitary.
(22) If Tn : H → H, n = 1, 2, 3, ... are normal linear operators and Tn → T ,
then show that T is normal.
(23) show that limit of a weakly convergent sequence is unique.
(24) Let {Tn } be a sequence of self adjoint linear operators Tn : H → H
on a Hilbert space H. If Tn → T , then show that T is self adjoint bounded
linear operator.
(25) Let X be a normed space and xn → x weakly. Show that every subsequence of {xn } converges weakly to x.
(26) Define weak and strong convergence in a normed space. Give an example to show that weak convergence does not imply strong convergence.
(27)Show that in a finite dimensional space weak convergence implies strong
convergence.
(28) If {xn } and {yn } are sequences in the same normed space, show that
xn → x weakly and yn → y weakly implies xn + yn → x + y weakly and
αxn → αx weakly, for all scalars α.
(29) Define compact linear operator and show that 0 operator on any
normed space is compact.
(30) When do you say that a normed space is reflexive. Which of the following spaces is not reflexive:
(a) Rn
(b) lp
(c) l1
(d) Any Hilbert space.