I need an example of a Banach space which is not a Hilbert space with proof.
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Check out Kreyzsig's Introductory Functional Analysis with Applications, page 133. Your example is $\displaystyle l^{p},$ with $\displaystyle p\not=2.$
To prove that a Banach space is not a Hilbert space, show that its norm does not satisfy the parallelogram identity.
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