Due November 6
Problem 1
1. Let $latex X$ be a real inner product space. Then, for $latex x,y\in X$,
$latex (x,y) = \dfrac{1}{4}\big( ||x+y||^2 - ||x-y||^2\big).$
2. Let $latex X$ be a complex inner product space. The, for $latex x,y\in X$,
$latex (x,y) = \dfrac{1}{4}\big( ||x+y||^2 - ||x-y||^2 + i||x+iy||^2 - i||x-iy||^2 \big).$
Problem 2
Let $latex (X,||\cdot||)$ be a normed space such that, for $latex x,y\in X$,
$latex ||x + y||^2 +||x - y||^2 = 2||x||^2 + 2||y||^2.$
Then $latex ||\cdot||$ is induced by an inner product.
Problem 3
Let $latex X$ be a finite dimensional inner product space over $latex \K$, say $latex \dim X = l$. Then $latex X$ is isometrically isomorphic to $latex \K^l$; i.e. there exists an isomorphism $latex \Phi:X\to\K^l$ such that
$latex (x,y) = (\Phi x, \Phi y)$
for all $latex x,y\in X$.
Problem 4
Let $latex Y$ be a closed subspace of the Hilbert space $latex X$, and define $latex T:X\to Y$ as
$latex Tx = \Proj_Y x$.
Then $latex T$ is continuous.
Problem 5
Let $latex Y$ be a closed subspace of the Hilbert space $latex $, and let
$latex Y^\perp = \{x\in X:x\perp Y\}.$
1. $latex Y^\perp$ is closed.
2. $latex X\cong Y\oplus Y^\perp$.
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