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## 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 $\bar X$.