Solution to 3D.18 wrongly assumes that V is finite-dimensional

[4DA]

In referenced 3.61 vector spaces are set to be finite dimensional.

We can't make use assumption for V from 3D.18

[celiopassos]

I agree. Do you have a solution?

I know how to solve it if we can take a basis of V, but how to do it if Axler avoids basis of infinite-dimensional stuff?

[4DA]

Define $\phi_v$: $F$ -> $V$ as $\phi_v(f) = fv$, for $v \in V$ and $f \in F$.

$\phi_v \in L(F, V)$, so we can define $\Phi$: $V$ -> $L(F, V)$ as $ \Phi(v) = \phi_v $

Then we can examine the mapping to show that $\Phi$ is invertible.