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Let $E=\mathcal{C}([0,1],\mathbb{R})$. For all $(f,g) \in E\times E$, we define : $$\langle f,g\rangle =\int_0^1f(t)g(t) \, dt.$$

  1. Show that the application $(f,g) \longmapsto <f,g>$ is an inner product on $E$. We define for all $f \in E$, $\|f\|=\sqrt{\langle f,f\rangle}$.
  2. Show that, for all $f \in E$, we have $\|f\| \leqslant \|f\|_\infty$ .
  3. Deduce, using Banach's isomorphism theorem, that the space $(E,\|\cdot\|)$ is not a Hilbert space.

The first two questions are easy but I got stuck in the third one (I can solve it by using Parallelogram law ).

I'm an undergraduate student so please can someone recommend me a reference (book -website -...) which contains exercises on normed spaces(Linear applications, Hahn-Banach and Banach-Steinhauss theorems..) and Hilbert spaces. Thank you in advance.

Michael Hardy
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M-S
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  • I don't think you can use parallelogram law, because parallelogram law only sees whether or not a norm $\lVert \bullet\rVert$ may be written as $\sqrt{\phi(x,x)}$ for some symmetric bilinear (or Hermitian sesquilinear) positive definite product $\phi$. But it is completely oblivious of completeness. –  Aug 26 '20 at 20:06
  • Thank you @Gae.S. we have $||f||=\sqrt{<f,f>}$ so we can use parallelogram law, isn't it? – M-S Aug 26 '20 at 20:15
  • I'm interested more by Banach's isomorphism theorem how we can use it here !! – M-S Aug 26 '20 at 20:16
  • Standard usage is $\langle f,f\rangle,$ not $<f,f>.$ I edited accordingly and also fixed some other typesetting solecisms. – Michael Hardy Aug 26 '20 at 21:59
  • Thank you for editing my question @MichaelHardy. – M-S Aug 26 '20 at 22:19

1 Answers1

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If $(E, \|\cdot\|)$ were a Hilbert space, or even a Banach space, the Banach isomorphism theorem would imply that the identity map $i$ from $(E, \|\cdot\|_\infty)$ to $(E, \|\cdot\|)$ is an isomorphism of Banach spaces; in particular, that $i^{-1}$ is bounded. This would mean there is a constant $C$ such that $\|f\|_\infty \le C \|f\|$ for all $f \in E$. But you can explicitly construct a function $f$ such that $\|f\| \le 1$ and $\|f\|_\infty > C$.

Nate Eldredge
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