Equality singular value $\sigma_1(A)=\sup_{\lVert x\rVert=1}\lVert Ax\rVert=\sup_{\lVert x\rVert =1,\lVert y\rVert=1}\langle Ax,y\rangle$

Question

I am trying to solve: $\sigma_1(A) = \sup\limits_{\lVert x \rVert = 1} \ \lVert A x \rVert = \sup\limits_{\lVert x \rVert = 1, \lVert y \rVert = 1} \langle Ax,y\rangle$ where $\sigma_1(A)$ is the largest singular value of $A$, $A$ being an $n \times p$ matrix.

The first equality is thanks to the decomposition $A = U \Sigma V^T$, with $U$ and $V$ being unitary, but I am not sure about the second one.

Thank you