1)Why embedding of ( not necessarily finite-dimensional) vector spaces $V\rightarrow W$ produces embedding of tensor algebras $T(V)\rightarrow T(W)$. 
I can prove it using Hamel basis in $W$ but is there a nicer ( more functorial ) argument?
2) How to prove the same statement for modules over an algebra instead of vector spaces?



Tensor algebra question
If $V$ is a subspace of $W$, consider the inclusion $f:V\to W$ and any map $g:W\to V$ such that $g\circ f=1\_V$; to construct $g$, you need to use bases or something equivalent, for it does not exist over, say, a general ring...


Now $T(-)$ is a functor, so $T(g)\circ T(f)=T(1\_V)=1\_{T(V)}$. It follows that the map $T(f):T(V)\to T(W)$ is injective.


