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

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

### Problem 3

Let Y be a close subspace of the Hilbert space X and $T:X\to Y$ 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. $X\cong Y\oplus Y^\perp$.