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.