4. (10 points) Let X be the normed linear space of all simple functions in L(E). Show that X is n...
please solve it by easy way , and send clear picture . 2. Let Cla,히 be the space of continuous functions and define l|-lla via Show that (Cla, b),Il a) is a normed linear space. Moreover, prove that (Cla. b,1la) is not a Banach space is a normed linear space. Moreover, prove that (Cla, b 2. Let Cla,히 be the space of continuous functions and define l|-lla via Show that (Cla, b),Il a) is a normed linear space. Moreover, prove...
3. [3 points) Let S T be bounded linear operators on a normed space X. Show that for every de p(S) np(T) one has RX(T) - RX(S) = RX(T)(S-T)RX(S). 4. [3 points) Let T be a linear operator on 12 defined by Tr - (E2, E1, E3, E1,85,...) (permutation of first two components). Find and classify the spectrum of T.
Let (X, 11. I be a normed vector space and let E C X be an n-dimensional subspace. (a) Prove that E is complete. (b) Prove that E is closed. (c) Prove that dim E* = n, where E* is the algebraic dual of E (the space of all linear functionals on E).
Request solve attached question from functional analysis E10) Let X be a normed linear space over C. Regarding X as a linear space over R, let u X R be a real linear functional. Prove that the function f : X C defined by E10) Let X be a normed linear space over C. Regarding X as a linear space over R. let u: X R be a real linear functional. Prove that the function f : X -C defined...
Consider the space V of continuous functions on (0, 1] with the 2-norm 12 J f2 We saw in class that V is an incomplete normed linear space. (a) For a continuous function p on [0, 1], define a linear map Mp: V-V by Mpf-pf. Show that Mp is bounded and calculate its norm. (b) Is A = (Mplp E C(0,1)) a Banach algebra? Note that B(V) is necessarily incomplete, so it is not enough to prove that A is...
3. Recall that R([0, 1]) is the normed linear space of integrable functions, with norm 1/2 Ils le = (150)Par)". Let (fn)nen be a sequence of functions in R, defined by 1<3 fn(x) = 1 VI V 0 < (a) Prove that (fn)nen is Cauchy. (b) Prove that (fn) does not converge in R([0, 1]). (Note: If it did, then what must the limit function be? Can this candidate function be in R?)
Topology C O, 1 and be the supremum norm (a) Prove that (X || |) is a Banach space. You can assume that (X, | |) is a normed vector space (over R) |f|0supE0.1 \5(x)|.| 4. Let X C (b) Show that || |o0 that the parallelogram identity fails.] on X is not induced by any inner product. Hint: Check for all E[0, 1]. Show that {gn}n>1 (0, 1] BI= {gE X |9||<1} is a compact (c) For every 2...
4. Problem 15.6.19. Let X be a normed vector space, and suppose that there exists a topological isomorphism A: X + (1. Prove that there exists a sequence {Xn}nen in X such that every vector x E X can be uniquely written as X = > Cn (2) Xn, where ) Cn(x)] < 0. n=1 Remark: Such a sequence is called an absolutely convergent Schauder basis for X. n=1
(b) 4 Let F: X Y be a linear map between two normed spaces. Prove that F is continuous at Ojf and only if F is uniformly continuous on X.