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Homework 14, Real Analysis

Due November 30

Problem 1

Let Y be a finite dimensional subspace of the Hilbert space X. Then Y is closed in X.

Problem 2

For each $latex n\in\Z$, let $latex e_n(x) = e^{2\pi i nx}$. Then

$latex \displaystyle \int_0^1 e_n(x) \overline{e_m(x)} dx = \begin{cases} 1 & n=m\\ 0 & n\not=m. \end{cases}$


Problem 3

Let Y be a close subspace of the Hilbert space X and $latex T:X\to Y$ the orthogonal projection onto Y,

$latex Tx = \text{Proj}_Y x.$

T is continuous.

Problem 4

Let Y be a closed subspace of the Hilbert space X, and let

$latex Y^\perp = \{ x\in X: x\perp Y\}.$

  1. $latex Y^\perp$ is a closed subspace of X.

  2. $latex X\cong Y\oplus Y^\perp$.