﻿ Functional Analysis代写 MT4515代写 – 天才代写

# Functional Analysis代写 MT4515代写

2023-05-13 15:55 星期六 所属： 作业代写,留学生作业代写-北美、澳洲、英国等靠谱代写 浏览：338

## MT4515 Functional Analysis – Class Test 2

Functional Analysis代写 Show that (T fn) is an equicontinuous family of functions. Why does (T fn) have a convergentsubsequence?

Answer all three questions. You should fully justify your answers which should be presented clearly and logically. Standard theorems may be used provided their use is clearly indicated.

### Q1. As usual, C[0, 1] denotes the space of continuous real valued functions on [0, 1] and the supremum norm || || ∞ is given by ||f||∞ = supt∈[0,1] |f(t)|.  Functional Analysis代写

Let T : (C[0, 1], || || ) (C[0, 1], || || ) be the linear operator defifined by

(a) Show that T is bounded and fifind the induced norm ||T|| of T.

(b) Let (fn) C[0, 1] be a sequence of functions such that ||fn|| 1 for all n N. Show that (T fn) is an equicontinuous family of functions. Why does (T fn) have a convergentsubsequence?6 marks

### Q2. Recall that l 1 is the (real) normed space of real sequences  Functional Analysis代写

which is complete under the norm

Let U : l 1 l 1 be the linear operator given by

U(x1, x2, x3, . . .) = (2x2, 2x3, 2x4, . . .).  Functional Analysis代写

(a) Show that U is bounded and fifind the induced norm ||U|| of U.

(b) Find the eigenvalues and corresponding eigenvectors of U. Hence fifind the spectrum σ (U) of U.6 marks

Q3. Let (X.〈 . 〉) be a real inner product space and let Y be a vector subspace of X. Let x ∈ X. Show that if y ∈ Y is such that ||x- y|| takes its minimum value (i.||x – y|| = infz∈y ||x -z|| then (x – y,w) = 0 for all w ∈ Y.

Hint: consider (x – y – λw, x – y – λw) for > λ R.