Homework 14, Real Analysis

Due date: November 10

Problem 1

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

Problem 2

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

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

Apply the Gram-Schmidt process to the sequence f_n(x) = x^n to obtain the orthonormal polynomials p_0, p_1, p_2, p_3, such that each 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 T:X\to Y the orthogonal projection onto Y,

Tx = \text{Proj}_Y x.

T is continuous.

Problem 4

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

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

  1. Y^\perp is a closed subspace of X.
  2. X\cong Y\oplus Y^\perp.

Problem 5

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

  1. \bar X is a Hilbert space.
  2. If X is separable, so is \bar X.
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