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## Due November 6

### Problem 1

1. Let $X$ be a real inner product space. Then, for $x,y\in X$,

$(x,y) = \dfrac{1}{4}\big( ||x+y||^2 - ||x-y||^2\big).$

2. Let $X$ be a complex inner product space. The, for $x,y\in X$,

$(x,y) = \dfrac{1}{4}\big( ||x+y||^2 - ||x-y||^2 + i||x+iy||^2 - i||x-iy||^2 \big).$

### Problem 2

Let $(X,||\cdot||)$ be a normed space such that, for $x,y\in X$,

$||x + y||^2 +||x - y||^2 = 2||x||^2 + 2||y||^2.$

Then $||\cdot||$ is induced by an inner product.

### Problem 3

Let $X$ be a finite dimensional inner product space over $\K$, say $\dim X = l$. Then $X$ is isometrically isomorphic to $\K^l$; i.e. there exists an isomorphism $\Phi:X\to\K^l$ such that

$(x,y) = (\Phi x, \Phi y)$

for all $x,y\in X$.

### Problem 4

Let $Y$ be a closed subspace of the Hilbert space $X$, and define $T:X\to Y$ as

$Tx = \Proj_Y x$.

Then $T$ is continuous.

### Problem 5

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

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

1. $Y^\perp$ is closed.

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