ASSIGNMENT II MTL 411 FUNCTIONAL ANALYSIS 1. For x = (x(1); x(2);::: ) 2 l1, let T : l1 ! l1 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 l1. 2. If T : X ! Y is a linear operator such that there exists c > 0 and 0 =6 x0 2 X satisfying kT xk ≤ ckxk 8 x 2 X; kT x0k = ckx0k; then show that T 2 B(X; Y ) and kT k = c. 3. For x = (x(1); x(2);::: ) 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 kSk = 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 kT k = 1 (vi) T is not injective. (vii) T is not an isometry. (viii) T is surjective. (ix) TS = I and ST =6 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 2 B(X; Y ) and S 2 B(Y; Z), the equality in the submulti- plicativity of the norms kS ◦ T k ≤ kSkkT k 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 kAxk = kxk for all x 2 X. Deduce that a linear isometry A is injective and kAk = 1. 6. Show with an example that an injective bounded linear map A with kAk = 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 Z 1 T x = x(t) dt: −∞ Is T a bounded linear map ? { P } 2 2 R1 1 2 1 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 X1 x = (x(1); x(2);:::; ) 7! a(i) x(i): i=1 Show that f is a bounded linear map and kfk = kak2. 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 k:k1, whereas it is a Banach space under the norm 0 kxk = kxk1 + kx k1: ( ) ( ) 1 1 Is the identity map I : C [0; 1]; k:k1 ! C [0; 1]; k:k 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) 8 x 2 X: 11. Let X and Y be normed linear spaces, X =6 0. Let F : X ! Y be a bounded linear map. Show that there exists fxng in X such that kxnk = 1 and kF (xn)k ! kF k as n ! 1: 12. Let S and T be operators on C[0; 1] defined as Z 1 Sx(t) = t x(u) du; T x(t) = tx(t); 8 t 2 [0; 1]: 0 (i) Prove that S and T are bounded linear maps. (ii) Do S and T commute ? (iii) Find kSk, kT k, kST k, and kTSk. 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) = x0(1); x 2 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. P1 Consider the linear map f(x) = j=1 x(j). Prove that f is continuous with respect to the norm k:k1 on C00, but discontinuous with respect to the norm k:k2: 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 =6 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 2 [1; 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 ≤ 1, let A : l ! l be defined by p Ax(i) = λix(i); i 2 N; x 2 l : k k j j Prove that A is a bounded linear operator and A = supn2N 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 2 X0 is a bounded linear operator, and kA0k ≤ kAk. (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 2 X0 and kAk = kA0k: R C k k 1 j j C k k 20. Let X1 = [0; 1] with norm x 1 := 0 x(t) dt and X2 = [0; 1] with norm x 1 := max0≤t≤1 jx(t)j. 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 discon- tinuous 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 Xr r kxk1 = jx(i)j; x = (x(1); x(2); : : : ; x(r)) 2 K : i=1 Let M = (k ) be an m × n matrix with scalar entries and i;j ( ) Xm α1 := max jki;j j : j = 1; 2; : : : ; n: ; i=1 that is, α1 is the maximum of the column sums of the matrix (jki;j j). Show that the operator norm of the linear operator TM defined by TM (x) = Mx satisfies kTM k = α1: 24. Let Km and Kn be endowed with the l1-norm defined by r kxk1 = maxfjx(i)j : i = 1; 2; : : : ; rg; x = (x(1); x(2); : : : ; x(r)) 2 K : Let M = (k ) be an m × n matrix with scalar entries and i;j 8 9 <Xn = j j α1 := max : ki;j : i = 1; 2; : : : ; m; ; j=1 that is, α1 is the maximum of the row sums of the matrix (jki;jj). Show that the operator norm of TM defined by TM (x) = Mx satisfies kTM k = α1: 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);::: ) 2 X P1 and each i = 1; 2;::: the series j=1 ki;j x(j) is convergent and if we let X1 Mx(i) = ki;jx(j); j=1 then Mx 2 Y . 4 MTL 411 FUNCTIONAL ANALYSIS 1 (i) Let X = l = Y . Consider the supremum of the column sums of the matrix (jki;jj), namely, ( ) X1 α1 := sup jki;jj : j = 1; 2;::: : i=1 1 1 Prove that if α1 < 1, then M defines a continuous linear map from l into l . (ii) Let X = l1 = Y . Consider the supremum of the row sums of the matrix (jk j), namely 8 9 i;j <X1 = j j α1 = sup : ki;j : i = 1; 2;::: ; : j=1 1 1 Prove that if α1 < 1, then M defines a continuous linear map from l into l . p (iii) Let X = l = Y , 1 < p < 1. Prove that if α1 < 1, α1 < 1, 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 2 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.