Extend Isometry from Semi-normed to Normed Space

[wayneckm]

Hello all,


May someone help me on this question:

Suppose the map F is an isometry which maps a dense set H of a semi-normed space \mathcal{H} to a normed space \mathcal{G}, now the theorem said we can extend this isometry in a unique manner to a linear isometry of the semi-normed space \mathcal{H}.

So I do not understand how and why can we do so?

Thanks very much!


Wayne

 

[wayneckm]

Is it because, due to the dense set, any element h in \mathcal{H} is the limit of a sequence of elements \{h_{n}\}\ in H, so this forms a Cauchy sequence, and then, under the isometric map, we can obtain a Cauchy sequence in \mathcal{G}, however, I think we should assume the space \mathcal{G} is complete and Hausdorff so that we are guaranteed there exists a unique limit, and so we can define such element as F(h) ?
Am I correct? Thanks.