Embedding vs continuous injection (in topological vector spaces) 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
 A: Think of the torus as the quotient of $\mathbb R^2$ in the usual way: $(x, y) \mapsto (x \bmod 1, y \bmod 1)$. 
Now look at the line $y = \pi x$ in the plane. Under this quotient, it becomes a curve in the torus, and the map is 1-1 and continuous, but locally, in the image, it doesn't look the way you expect, i.e., like a line in $\mathbb R^2$, where near the line there is "no other stuff". 
A: As $X$ and $f(X)$ are not homeomorphic, if we identify $X$ with $f(X)$, the topology on $X$ is not the same as the subspace topology, so that one has to deal with the topology on $X$ seperately. This is important e.g. in the difference between immersed and embedded submanifolds.
A: Both refer to induced structures...
Subspace* and embedding:
$$\iota:S\hookrightarrow\Omega:\quad\mathcal{T}(S)=\{U\subseteq S:U\in\iota^{-1}\mathcal{T}(\Omega)\}$$
(The embedding being a homeomorphism onto its image.)
Projection and quotient space:
$$\pi:\Omega\twoheadrightarrow\tilde\Omega:\quad\pi^{-1}\mathcal{T}(\tilde\Omega)=\{\tilde{U}\subseteq\tilde\Omega:\pi^{-1}\tilde{U}\in\mathcal{T}(\Omega)\}$$
(The projection being an open continuous map.)
...There are many more considerable structures.
*Subspace in the general sense: $S\nsubseteq\Omega$
