how to prove translation invarient metric on vector space makes it topological vector space? Topological vector space has a property that addition and scalar multiplication are continuous operations.
I think it obvious that metric (translation invarient) on vector space makes it topological vector space But how to prove both operation continuous . I do not have an idea.
Please can anyone give me some hint or counterexample
Thanks a lot
 A: A translation invariant compatible metric implies that if addition is continuous at some point (typically $(0,0)$, of course) it is continuous everywhere. But it does not imply that it is continuous at some point, that is what being a TVS means. Also it means nothing for scalar multiplication: the discrete topology/metric on $X=\Bbb R$ makes addition continuous and the metric is translation invariant (always $1$, basically), but scalar multiplication as a map from $X \times (\Bbb R, \mathcal{T}_e) \to X$ (where the scalars $\Bbb R$ carries the usual Euclidean topology !) is not continuous, as $(1, \frac1n) \to (1,0)$ while $\frac1n \cdot 1 = \frac1n \not\to 0= 1\cdot 0$ in $X$. 
A Fréchet space is then assumed to be locally convex to make up for the scalar misbehaviour (we can use seminorms to build a better behaved compatible metric).
So we really need for Fréchet spaces that we have a TVS and a complete metric plus local convexity, e.g. We cannot replace TVS with a tarnslation invariant metric alone...
