Let $E, E^*$ be a pair of dual spaces and assume that $\Phi: E^* \times E \to \Gamma$ is a bilinear function such that $$\Phi(\tau^{*-1}x^*, Tx) = \Phi(x^*, x)$$ for every pair of dual automorphisms. Prove that $\Phi(x^*, x) = A<x^*, x>$ where $A \in \Gamma$.
First of all, I'm not sure what $<x^*, x>$ means. I assumed it is equal to $x^*(x)$ since $x^*$ is a linear functional and Inner Product is only defined between vectors of the same space. Secondly, by taking $x^*$ and $x$ to be basis vectors and $\tau$ be the automorphism which swaps a specific choice of basis vectors or just scales a single vector, I was able to prove this. However, I was wondering if it is possible to do so without basis? (That is, prove this for vector spaces of infinite dimensions).