Let $A \in \mathbb{R}^{m \times n}$ be any real matrix and let $V \in \mathbb{R}^{n \times d}$ be a matrix whose columns are orthonormal.
Is it true that $$ \left| \left| A\right| \right|_1 \leq \left| \left| AV\right| \right|_{1,1}?$$
If so, how can it be proved?
Please advise.
P.s. the $L_{p,q}$-norm can be found Here, and $\left| \left| A \right| \right|_1 = \sup_{\left| \left| x \right| \right|_1 = 1} \left| \left| Ax\right| \right|_1$.