[Math] Existence of a linear transformation in an infinite dimension vector space.

Question

If $V$ and $W$ are vector spaces, $\beta=\{v_1, \ldots , v_n\}$ is a finite a basis for $V$ and $\{w_1, \ldots , w_n\}\subset W$, we know there is an unique linear transformation $T:V\rightarrow W$ such that $T(v_i)=w_i$ for $i=1, 2, \ldots , n$
Is this valid when $V$ is not finite-dimensional?

Answer 1

The fact that linear maps are uniquely determined by their restriction to a basis of the source vector space is general, and not restricted to the finite dimensional case. Of course if $V$ is infinite dimensional, any basis of $V$ will be infinite, and the family of their images $w_\alpha$ that must be specified to determine $T$ will be infinite as well.