So on the german wikipedia page of topological vector spaces it is written that, if a topological vector space is a Hausdorff space, then translation by a vector and dilation by a scalar are homeomorphisms. However, I have looked at several lecture notes and it is always stated (and proofed) that these two mappings are homeomorphism for general topological vector spaces (even those that are not Hausdorff). So I wanted to confirm that it is true that for any - possibly non Hausdorff - topological vector space the translation and dilation are homeomorhpisms.
Also, can anybody think of a reason why on the german wikipedia they have restricted that statement to topological vector spaces with the Hausdroff property?
By definition translation by a vector v and multiplying by a scalar c are continuous. They are also bijective, because translation by -v and scaling by 1/c are inverses. Again by definition they are continuous wich proofs your question. No Hausdorffness needed.
Some authors require the spaces to be Hausdorff since it gives us uniqueness of limits for example.