Nondegenerate dual pairings inducing isomorphisms

Question

Given two $k$-vector spaces $A$ and $B$ and a bilinear pairing $\langle-,-\rangle: A \times B \to k$ that is non-degenerate in both entries, do we necessarily have a linear isomorphism between the two?

Answer 1

A linear isomorphism between two $k$-vector spaces $A, B$ exists if and only if $\dim_kA=\dim_kB$ (as cardinal numbers).

If either of $A$ or $B$ has finite dimension, then the other one must have the same dimension. In particular, in this case the answer to your question is yes. Namely, say $\dim_k A =n \in \mathbb N_0$. Then it is known that $\dim_k A^* =n$ as well, and since non-degeneracy via comment gives an injection $B \hookrightarrow A^*$, we get $\dim_k B \le n$. On the other hand, the injection $A \hookrightarrow B^*$ then implies $n \le \dim_k B^* = \dim_k B$.

However, if both vector spaces have infinite dimension, there are counterexamples. Namely, for $A=\bigoplus_{n\in \mathbb N} k$ (countable direct sum), the dual vector space is $A^*=\prod_{n\in \mathbb N} k$ whose dimension is uncountably infinite. However, setting $B=A^*$ still induces a non-degenerate pairing by duality. -- In general, for infinite dimensional vector spaces, the dual is not isomorphic to (has dimension of bigger cardinality than) the original space, in spite of an obvious non-degenerate pairing between the two.