Let $A\in\mathbb{R}^{m\times n}$ be a block matrix which is partitioned into submatrices $A_{ij} \in \mathbb{R}^{m_i\times n_j}$. Let $\|\cdot\|_{p\times q}$ denote the spectral (operator) norm on $p\times q$ matrices. Is it true that
$$\|A\|_{m\times n} \leq \sum_{i,j}\|A_{ij}\|_{m_i \times n_i}\:?$$
Let $A^{ij}$ denote the $m \times n$ matrix with the sub-matrix $A_{ij}$ at the appropriate position and with the rest of the entries as zero. Now note that $$A = \sum_{i,j}A^{ij}$$ Also, note that due to triangle inequality that every norm has to obey (including spectral norm) $$\|A\| \leq \sum_{i,j}\big\|A^{ij}\big\|$$ Now $\|A^{ij}\|_{m\times n}=\|A_{ij}\|_{m_i\times n_j}$ (can you verify that? essentially boils down to proving that padding zeros on the "borders" of a matrix doesn't change spectral norm. Hint: Use SVD) and your required result follows.