# Banach space & Hilbert space

#### manik

I need an example of a Banach space which is not a Hilbert space with proof.

#### Ackbeet

Check out Kreyzsig's Introductory Functional Analysis with Applications, page 133. Your example is $\displaystyle l^{p},$ with $\displaystyle p\not=2.$

#### Opalg

To prove that a Banach space is not a Hilbert space, show that its norm does not satisfy the parallelogram identity.

Ackbeet