Let $X,Y$ be Hilbert spaces, $X$ continuously and densely embedded into $Y$, and $(u_k)_k$ be a complete orthogonal system in $V$ and an orthonormal basis of $Y$ s.t. $\langle u_k,u_l\rangle_X=\lambda_k\delta_{kl}$ and $\langle u_k,u_l\rangle_Y=\delta_{kl}$ for some $0<\lambda_1\le\lambda_2\le\cdots\to\infty$.
If $v\in X$, we can show $\left\|v\right\|_Y=\left(\sum_{i=1}^\infty c_i^2\right)^{1/2}$ and $\left\|v\right\|_X=\left(\sum_{i=1}^\infty\lambda_ic_i^2\right)^{1/2}$, where $c_i:=\langle v,u_i\rangle_Y$.
Are we able to conclude that the embedding of $X$ into $Y$ is compact?
If $u_k$ is a complete orthogonal system in $X$ and an orthonormal basis in $Y$ with $\langle u_k,u_l \rangle_X =\lambda_k \delta_{j,k}$, then $\|u_k\|=\sqrt{\lambda_k}$ so that $e_k=\lambda_k^{-1/2} u_k$ is an orthonormal basis of $X$. Identifying $\ell^2 \cong X$ (mapping the standard unit sequence to $e_k$) and $Y\cong \ell^2$ (mapping $u_k$ to standard unit sequence), the inclusion of $X$ into $Y$ becomes the diagonal operator $\ell^2\to\ell^2$ mapping a sequence $(x_k)_k$ to $(\lambda_k^{-1/2} x_k)_k$. This is compact if and only of $\lambda_k^{-1/2}\to 0$ (i.e., $\lambda_k\to\infty$) and it is nuclear if and only if $\sum_k |\lambda_k^{-1/2}|<\infty$ by Grothendieck-Pietsch.