L∞ , Let T : L ∞ → L∞ Be Defined by Tx = ( X(1), X(2) 2 , X(3) 3 ,... ) Pr

ASSIGNMENT II

1. For x = (x(1), x(2),... ) ∈ l∞, let T : l∞ → l∞ be deﬁned by ( ) x(2) x(3) T x = x(1), , ,... 2 3 Prove that (i) T is a bounded linear operator (ii) T is injective (iii) Range of T is not a closed subspace of l∞.

2. If T : X → Y is a linear operator such that there exists c > 0 and 0 ≠ x0 ∈ X satisfying

∥T x∥ ≤ c∥x∥ ∀ x ∈ X; ∥T x0∥ = c∥x0∥, then show that T ∈ B(X,Y ) and ∥T ∥ = c.

3. For x = (x(1), x(2),... ) ∈ l2, consider the right shift operator S : l2 → l2 deﬁned by Sx = (0, x(1), x(2),... ) and the left shift operator T : l2 → l2 deﬁned by T x = (x(2), x(3),... ) Prove that

(i) S is a abounded linear operator and ∥S∥ = 1. (ii) S is injective. (iii) S is, in fact, an isometry. (iv) S is not surjective. (v) T is a bounded linear operator and ∥T ∥ = 1 (vi) T is not injective. (vii) T is not an isometry. (viii) T is surjective. (ix) TS = I and ST ≠ I. That is, neither S nor T is invertible, however, S has a left inverse and T has a right inverse.

Note that item (ix) illustrates the fact that the Banach algebra B(X) is not in general commutative.

4. Show with an example that for T ∈ B(X,Y ) and S ∈ B(Y,Z), the equality in the submulti- plicativity of the norms ∥S ◦ T ∥ ≤ ∥S∥∥T ∥ may not hold.

5. Let X and Y be normed linear spaces and A : X → Y be a linear map. Show that A is an isometry if and only if ∥Ax∥ = ∥x∥ for all x ∈ X. Deduce that a linear isometry A is injective and ∥A∥ = 1.

6. Show with an example that an injective bounded linear map A with ∥A∥ = 1 need not be an isometry. 1 2 MTL 411 FUNCTIONAL ANALYSIS

7. Let X be the space of all continuous functions f on R that vanish outside a finite interval [a, b] (depending on f). Endow X with the sup-norm. Define T : X → R by ∫ ∞ T x = x(t) dt. −∞ Is T a bounded linear map ? { ∑ } 2 ∈ R∞ ∞ 2 ∞ ∈ 2 8. Let l := x = (x(1), x(2),... ) : i=1(x(i)) < and a l be a fixed non zero element. Define f : l2 → K by ∑∞ x = (x(1), x(2),..., ) 7→ a(i) x(i). i=1

Show that f is a bounded linear map and ∥f∥ = ∥a∥2.

9. Let C1[0, 1] denote the space of all continuously diﬀerentiable function x deﬁned on [0, 1]. 1 Then C [0, 1] is an incomplete normed linear space under the supnorm ∥.∥∞, whereas it is a Banach space under the norm ′ ∥x∥ = ∥x∥∞ + ∥x ∥∞. ( ) ( ) 1 1 Is the identity map I : C [0, 1], ∥.∥∞ → C [0, 1], ∥.∥ bounded ?

10. Let f1 and f2 be two linear functionals on a linear space X. Show that if they have the same nullspace, then there is a nonzero scalar k such that

f2(x) = kf1(x) ∀ x ∈ X.

11. Let X and Y be normed linear spaces, X ≠ 0. Let F : X → Y be a bounded linear map. Show that there exists {xn} in X such that ∥xn∥ = 1 and ∥F (xn)∥ → ∥F ∥ as n → ∞.

12. Let S and T be operators on C[0, 1] deﬁned as ∫ 1 Sx(t) = t x(u) du, T x(t) = tx(t), ∀ t ∈ [0, 1]. 0 (i) Prove that S and T are bounded linear maps. (ii) Do S and T commute ? (iii) Find ∥S∥, ∥T ∥, ∥ST ∥, and ∥TS∥.

13. Let X = C1[0, 1], the space of all continuously diﬀerentiable scalar-valued functions on [0, 1] with the sup norm. Deﬁne f : X → K by f(x) = x′(1), x ∈ X. Is f a continuous linear functional ? Is N(f) a closed subspace of X ?

14. Let X = C00, the space of all scalar sequences which have only ﬁnitely many nonzero terms. ∑∞ Consider the linear map f(x) = j=1 x(j). Prove that f is continuous with respect to the norm ∥.∥1 on C00, but discontinuous with respect to the norm ∥.∥2.

15. Prove that the null space of a discontinuous linear functional on a normed linear space X is dense in X.

16. Prove that a closed linear functional is always continuous. Consider the map f : R → R deﬁned by { 1 if x ≠ 0, f(x) = x 0 if x = 0. Show that f is closed but not a continuous map on R. Does this contradict the ﬁrst assertion given in the question ? ASSIGNMENT II 3

17. Let p, r ∈ [1, ∞] and p ≤ r. Prove that the inclusion operator I : lp → lr is a bounded linear operator.

p p 18. Let (λn) be a bounded sequence in K. For 1 ≤ p ≤ ∞, let A : l → l be deﬁned by p Ax(i) = λix(i), i ∈ N, x ∈ l . ∥ ∥ | | Prove that A is a bounded linear operator and A = supn∈N λn .

19. Let X and Y be normed linear spaces and X0 be a subspace of X. Prove the following: (i) If A : X → Y is a bounded linear operator, then the restriction A0 : X0 → Y deﬁned by A0x = Ax for all x ∈ X0 is a bounded linear operator, and ∥A0∥ ≤ ∥A∥. (ii) Suppose A0 : X0 → Y is a bounded linear operator, X0 is dense in X and Y is a Banach space. Then A0 has a unique norm-preserving bounded linear extension to all of X, i.e., there exists a unique bounded linear operator A : X → Y such that Ax = A0x for all x ∈ X0 and ∥A∥ = ∥A0∥. ∫ C ∥ ∥ 1 | | C ∥ ∥ 20. Let X1 = [0, 1] with norm x 1 := 0 x(t) dt and X2 = [0, 1] with norm x ∞ := max0≤t≤1 |x(t)|. Then the identity operator from X1 to X2 is not continuous - Why?

21. Show that, on every inﬁnite dimensional normed liner space, there exists at least one discontinuous linear functional.

22. Let M = (ki,j) be an m × n matrix with scalar entries. Then M deﬁnes a linear operator n m TM : K → K deﬁned by TM (x) = Mx. Is TM a bounded linear map ?

23. Let Km and Kn be endowed with the l1-norm deﬁned by ∑r r ∥x∥1 = |x(i)|, x = (x(1), x(2), . . . , x(r)) ∈ K . i=1 Let M = (k ) be an m × n matrix with scalar entries and i,j { } ∑m α1 := max |ki,j | : j = 1, 2, . . . , n. , i=1

that is, α1 is the maximum of the column sums of the matrix (|ki,j |). Show that the operator norm of the linear operator TM deﬁned by TM (x) = Mx satisﬁes

∥TM ∥ = α1.

24. Let Km and Kn be endowed with the l∞-norm deﬁned by r ∥x∥∞ = max{|x(i)| : i = 1, 2, . . . , r}, x = (x(1), x(2), . . . , x(r)) ∈ K . Let M = (k ) be an m × n matrix with scalar entries and i,j   ∑n  | | α∞ := max  ki,j : i = 1, 2, . . . , m , j=1

that is, α∞ is the maximum of the row sums of the matrix (|ki,j|). Show that the operator norm of TM deﬁned by TM (x) = Mx satisﬁes

∥TM ∥ = α∞.

25. Consider an inﬁnite matrix M = (ki,j) with scalar entries. We say that M deﬁnes a linear map from a sequence space X to a sequence space Y if for every x = (x(1), x(2),... ) ∈ X ∑∞ and each i = 1, 2,... the series j=1 ki,j x(j) is convergent and if we let ∑∞ Mx(i) = ki,jx(j), j=1 then Mx ∈ Y . 4 MTL 411 FUNCTIONAL ANALYSIS

1 (i) Let X = l = Y . Consider the supremum of the column sums of the matrix (|ki,j|), namely, { } ∑∞ α1 := sup |ki,j| : j = 1, 2,... . i=1 1 1 Prove that if α1 < ∞, then M defines a continuous linear map from l into l . (ii) Let X = l∞ = Y . Consider the supremum of the row sums of the matrix (|k |), namely   i,j ∑∞  | | α∞ = sup  ki,j : i = 1, 2,...  . j=1 ∞ ∞ Prove that if α∞ < ∞, then M defines a continuous linear map from l into l . p (iii) Let X = l = Y , 1 < p < ∞. Prove that if α1 < ∞, α∞ < ∞, then M defines a continuous linear map from lp to lp.

26. If X0 is a subspace of a normed linear space X, then the inclusion operator I0 : X0 → X, i.e., I0x = x for all x ∈ X0, is a compact operator if and only if X0 is ﬁnite dimensional.

27. The space C (Rn, Rm) of all compact linear maps from Rn into Rm can be identiﬁed with the space of all m × n matrices - Why ?

28. Give examples of a commutative unital Banach algebra, commutative Banach algebra without unit element, and noncommutative unital Banach algebra.

29. Show that the right-shift operator and the left shift operator on lp are not compact operators.

30. Show with an example that inverse of a bijective bounded linear map need not be bounded.