When working with topological vector spaces (say $X,Y$), the term “embedding” is often used for a continuous injection $f:X\rightarrow Y$.
Now, $f$ is of course a bijection onto its image, but it's not necessarily a homeomorphism (as we would normally require of an embedding in topology).
Can anyone explain to me exactly what we're loosing by not having $X$ and $f(X)$ homeomorphic? And is there something special about linear spaces that makes whatever we're missing (more or less) irrelevant?
I suppose that the subspace topology on $f(X)$ must in some sense be coarser than the topology on $X$ if the inverse of $f$ (defined on the range of $f$ ) fails to be continuous