Please give me a complete proof that a Banach space s may not be a Hilbert space. Solve example which is a Banach space but not Hilbert space
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1Welcome to the community! Please take a look at How to ask a good question to avoid down votes. – Ѕᴀᴀᴅ May 20 '18 at 11:42
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Possible duplicate of A Banach space that is not a Hilbert space – Alex Vong May 20 '18 at 13:06
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Towards the second part of your question. Consider the set of all continuous functions on a compact interval $[a,b]$ for $a < b$ and $a,b \in \mathbb{R}$, i.e. $f: [a,b] \to \mathbb{R}$. And the uniform norm, $\lVert \cdot\rVert_\infty$. Then this set together with this norm, i.e. $(C([a,b]),\lVert \cdot\rVert_\infty)$, is a Banach space but not a Hilbert space as the uniform norm is not induced by a scalar product.
LenC
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Another example is the $l^p$ space for $p \neq 2$. This follows easily from the parallelogram law.
eddie
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