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Homework 14, Real Analysis

Due date: November 10

Problem 1

Let Y be a vector subspace of the normed space X. Then its closure $latex \bar Y$ is also a vector subspace of X.

Problem 2

Let $latex X=C([-1,1])$ with the inner product

$latex \displaystyle (f,g) = \int_{-1}^1 f\bar g.$

Apply the Gram-Schmidt process to the sequence $latex f_n(x) = x^n$ to obtain the orthonormal polynomials $latex p_0, p_1, p_2, p_3$, such that each $latex p_n$ is of degree n.

These are the first Legendre polynomials.

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

Problem 5

Let X be an inner product space and $latex \bar X$ its completion.

  1. $latex \bar X$ is a Hilbert space.

  2. If X is separable, so is $latex \bar X$.