Given $A \in \mathbb{R}^{p \times d}$, $X \in \mathbb{R}^{p \times r}$ and $Y \in \mathbb{R}^{d \times r}$, where each of $X$ and $Y$ has orthonormal columns and $r=\operatorname{rank}(A)\leq \min{(p, d)}$, prove that $|\langle X, AY \rangle| \leq \sqrt{r\sum_{i=1}^r \sigma_i^2(A)}$, where $\sigma_i(A)$ is the $i$th largest singular value of $A$.
I've rewritten $\sum_{i=1}^r \sigma_i^2(A)$ as $\Vert A \Vert_F$, and rearranged the expression, although I'm not quite sure what to do, and don't know if I'm even going in the right direction.
By changing two orthonormal bases, you may assume that $A$ is a singular value matrix. It follows that $$ |\langle X,AY\rangle|=\sum_{k=1}^r\sigma_i(A)\langle X_{\ast k},Y_{\ast k}\rangle\le\sum_{k=1}^r\sigma_i(A). $$ Let $s=(\sigma_1(A),\ldots,\sigma_r(A))^T$ and $e=(1,\ldots,1)^T$. Then $$ \sum_{k=1}^r\sigma_i(A)=\langle s,e\rangle\le\|s\|_2\|e\|_2=\sqrt{r\sum_{i=1}^r\sigma_i^2(A)}. $$