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Show that a norm on an inner product space satisfies parallelogram law. Hence use the parallelogram law to show that the space of continuous real functions defined on the interval $[a,b]$ is not a Hilbert space. Here I did the first part. Let $X$ be an inner product space and $x,y\in X$ consider $\Vert x+y\Vert ^2$ and $\Vert x-y\Vert ^2$ Then on adding I get $\Vert x+y\Vert ^2+\Vert x-y\Vert ^2=2(\Vert x\Vert ^2 +\Vert y\Vert ^2)$ For the second part I don’t know how can a parallelogram law prove $C[a,b]$ is not a Hilbert space, help me please.

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If $C([a,b])$ were a Hilbert space then the parallelogram law would hold. Show it does not. – user20266 May 27 '12 at 11:28

Consider functions $$ x(t)=\cos\left(\frac{\pi}{2}\frac{t-a}{b-a}\right)\qquad y(t)=\sin\left(\frac{\pi}{2}\frac{t-a}{b-a}\right)\qquad $$ then $$ \Vert x+y\Vert=\sqrt{2},\qquad\Vert x-y\Vert=1\qquad \Vert x\Vert=\Vert y\Vert=1 $$

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