Tarea 13, Análisis real

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