I am Referring to the book Linear Algebra by Hoffman and Kunze $2$e, page $84$ section $3.3$.
It defines Isomorphism as follows:
If $V$ and $W$ are vector spaces over a field $\mathbb{F}$, then any one-one $T$ $\in$ $\mathcal{L}(V, W)$ of $V$ onto $W$ is called an Isomorphism from $V$ to $W$.
Standing alone, the definition seems correct, but the authors go on to say that:
if $V$ is Isomorphic to $W$ via an Isomorphism $T$, then $W$ is Isomorphic to $V$ because $T^{-1}$ is an Isomorphism from $W$ to $V$.
I feel this is erroneous, because as per the definition given $T$ is one-one, which doesn't imply the existence of an inverse on its own.
To correct it, either, the term "bijective" must be used instead of one-one, or there should be an additional condition that $~\dim(V) = \dim(W).$
Am I correct in this interpretation? Or am I missing something out?
I tried looking up in the various errata for this book, but couldn't find anything related to this.
Thanks.
Note that in the definition of isomorphism given by the author, $T$ is onto. So $T$ is bijective by definition.
Additional information: In general, people use the term into when they describe a map from a set $A$ into $B$. So onto is used for a specific purpose.