1) I know that all inner product space is also a normed space with the norm induce by the scalar product, but is the reciprocal true ? I mean, is all normed space also a inner product space ?
2) I know that all normed space is a metric space with the metric induced by the norm. Is the reciprocal true ? I mean, is all metric space also a normed space ?
Best Answer
#1: No. Norms which are induced by inner products are exactly those satisfying the "parallelogram law": $2 \| x \|^2 + 2 \| y \|^2 = \| x+y \|^2 + \| x-y \|^2$. In this case you have an inner product defined by the "polarization identity" $\langle x,y \rangle = \frac{1}{4} \left ( \| x+y \|^2 - \| x - y \|^2 \right )$ (with some small change in the complex case). Two norms on $\mathbb{R}^n$ for $n>1$ that do not have this property are the $1$ norm, $\| x \|_1=\sum_{i=1}^n |x_i|$, and the $\infty$ norm, $\| x \|_\infty = \max_i |x_i|$.
#2: No, there are many metric spaces which are not normed spaces. Many of these are not even vector spaces, but we can even equip vector spaces with metrics which are not compatible with norms (in the sense that there is no norm such that $\| x \|=d(x,0)$). For example, if we equip any vector space with positive dimension with the discrete metric, then the homogeneity property of the norm is violated: if $x \neq 0,\alpha \neq 0,|\alpha| \neq 1$, then $d(\alpha x,0)=d(x,0)=1 \neq |\alpha|$.