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Consider sequence spaces of the form,

$$\ell^p=\Big\{x=\left(x_j\right)_{j=1}^\infty \mathrel{}\big|\mathrel{} \sum_{j=1}^\infty \left\lvert x_j \right\rvert ^p\lt\infty\Big\}$$

for $1\le p\lt\infty$, equipped with the norm,

$$\left\|x\right\|_p:=\left(\sum_{j=1}^\infty\left\lvert x_j \right\rvert ^p\right)^{\frac{1}{p}}$$

for $x\in\ell^p$.

Now, in the case that $n=2$, the familiar norm on $\ell^2$ is obtained from the inner product, that is to say that the norm is induced by the inner product. And since $\ell^2$ is a infinite-dimensional Banach space, it follows that it is in fact an infinite dimensional Hilbert space.

Why is it, that when $p\neq2$, the space $\ell^p$ is not an inner product space?

Vlad
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Jeremy Jeffrey James
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1 Answers1

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For the case $p\neq 2$ the famous parallelogram law dos not hold so the space $l^p$ is not Hilbert. It is a theorem in many functional analysis text book such as Erwin Kreszig (Intoductory functional analysis with application)

Ali
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