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How can I prove that seminorms are convex?

Another question is that, when we talk about topological vector spaces, why do we emphasise neighborhoods about $0$? As far as I know, if we know the neighborhoods about $0$ then we can find all the open sets by translation. But can I not talk about open sets about any other point? I think I am missing something here. Thank you for helping.

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up vote 2 down vote accepted

Just use the axioms of a semi-norm: if $x,y$ are elements of the space and $\alpha\in [0,1]$, use triangular inequality and the fact that $\alpha$, $1-\alpha$ are non-negative.

If $V$ is a neighborhood of $0$, $x+V:=\{x+v,v\in V\}$ is a neighborhood of $x$. Since in the context of topological vector spaces translations are supposed to be continuous, we just focus on open sets which contain $0$ (we can deduce the other by translation).

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thank you for helping always. – Theorem Jun 5 '12 at 22:08
You are welcome. – Davide Giraudo Jun 5 '12 at 22:08
since $||x\alpha +(1-\alpha)x||\le ||x||$ , is it true that all normed spaces are convex ? I am sure that i am not understanding something here . – Theorem Jun 6 '12 at 8:53

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