If $u,v\in V$ then there is a linear transformation $T:V\to V$ such that $T(u)=v$. Suppose $V$ is a vector space, not necessarily finite dimensional. If $u,v\in V$, is there a linear transformation $T:V\to V$ such that $T(u)=v$. 

I proved the result for the finite dimensional case where I used a basis for $V$. What should one do for the infinite dimensional case?
Not always. If $u=0$, then $v$ must be $0$.

If $u\ne0$ and $V$ has an inner product, then $\displaystyle T(x)=\dfrac{\langle x,u\rangle}{\langle u,u\rangle}v$ works.

Geometrically, $T$ projects $V$ onto the subspace generated by $u$, which is canonically isomorphic to the subspace generated by $v$.
Not always. If $u=0$, then $v$ must be $0$.

If $u\ne0$ and $V$ has an inner product, then $\displaystyle T(x)=\dfrac{\langle x,u\rangle}{\langle u,u\rangle}v$ works.

Geometrically, $T$ projects $V$ onto the subspace generated by $u$, which is canonically isomorphic to the subspace generated by $v$.