In Wikipedia, bilinear mapping to the underlying field $f \in \operatorname{Hom}(V \otimes V, k)$ is defined as non-degenerate iff $X \mapsto f(-,X)$ is an isomorphism to the continuous (?) dual.
I tried to prove the proposition that $X \mapsto f(-,X)$ is an iso iff $X \mapsto f(X,-)$ is an iso, but only managed to prove that this proposition is equivalent to the proposition that $\varphi := \lbrace(f(-,X), f(X,-))\rbrace$ is an automorphism of $V^*$ (and a mapping, of course).
Is the proposition true? I assume it's likely because there is no mention of left vs. right degeneracy on the wiki.
I'm asking because I'm trying to study orthogonal complements in the most general setting, so I wanted to see what conditions on $f$ are required for it to define complement, when $V \cong U \oplus U^\perp$ and when left and right complements coincide.
EDIT: I am specifically concerned about the infinite-dimensional case. For the finite-dimensional case, the proposition is easy to prove using a variety of strategies (e.g. via determinants).