Is all normed space also inner product space? 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 ?

A norm is induced b yan inner product iff it satisfies the paralellogram law
$$2||u||^2 + 2||v||^2 = ||u -v||^2 + ||u+v||^2$$
And e.g. the supremum norm on $\mathbb{R}^2$ already fails this.
Even if we have a metric topological vector space with a translation invariant metric $(V,d)$ which is moreover complete (a so-called Fréchet space), so we have compatibility with the linear operations, $d$ need not be induced by a norm. 
A normed space has the property that every open neighbourhood $U$ of $0$ is bounded (in the sense that $\forall x, \exists t \in \mathbb{R}: tx \in U$, while this fails for many Fréchet spaces like $\mathbb{R}^\mathbb{N}$.
The $\ell^p$ spaces for $0 < p < 1$ fail normability as well, for other reasons (not locally convex), even though they have a nice metric structure.
