# Connexions

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# Banach and Hilbert Spaces

Module by: Richard Baraniuk. E-mail the author

A normed vector space is a vector space VV equipped with a norm ˙ ˙ .

## Example 1

In RN N with l 2 l 2 -norm: RN N is the space set, and l 2 l 2 -norm is the distance measure.

In RN N with l 1 l 1 -norm: RN N is the space set, and l 1 l 1 -norm is the distance measure.

RN N with l p l p -norm is the most general.

We call a NVS a Banach space. A Hilbert space is a Banach space equipped with an inner product < ˙ , ˙ > < ˙ , ˙ > .

Fundamental: The inner product generates the norm for the Hilbert space.

## Example 2

RN N with ( x , y ) =yTx ( x , y ) y x inner product, generates norm: x2= ( x , x ) =xTx x 2 ( x , x ) x x .

RN N with l 2 l 2 norm is a Hilbert space.

## Fact:

RN N with l p l p norm, p2 p 2 , is not a Hilbert space. There exists not inner product ( ˙ , ˙ ) ( ˙ , ˙ ) that can generate and l p l p norm p2 p 2
RN N with l 2 l 2 norm is a very special case:

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