1.3 Banach and hilbert spaces

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A normed vector space is a vector space $V$ equipped with a norm .

In $N$ with $l_{2}$ -norm: $N$ is the space set, and $l_{2}$ -norm is the distance measure.

In $N$ with $l_{1}$ -norm: $N$ is the space set, and $l_{1}$ -norm is the distance measure.

$N$ with $l_{p}$ -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.

$N$ with $(x, y) y x$ inner product, generates norm: $x 2 (x, x) x x$ .

$N$ with $l_{2}$ norm is a Hilbert space.

N

with

l^{p}

norm,

p 2

, is not a Hilbert space. There exists not inner product

(,)

that can generate and

l_{p}

norm

p 2

N

with

l_{2}

norm is a very special case:

Banach Spaces (N.V.S.)

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Source: OpenStax, Ee textbook. OpenStax CNX. Feb 13, 2009 Download for free at http://cnx.org/content/col10643/1.1

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