I need to show that if $\bf{M}$ is an $m \times n$ matrix, then $||\bf{M}(\bf{h})||\leq \alpha ||\bf{h}||$ for some $\alpha$ by estimating the entries of $\bf{M}$. The hint I am given is that $\bf{M}(\bf{h}) = \bf{y} = (y_1, \ldots, y_m)$ where $y_i=\bf{m}_i \cdot \bf{h}$ for some $\bf{m}_i \in R^n$ and apply the CS inequality.
2. If we denote by $(a_{k,j})_{1\leq k\leq m,1\leq j\leq n}$ the coefficients of the matrix $M$, we have to notice that $y_i =\sum_{j=1}^na_{ij}h_j$. Now, the question is: what is the vector $m_i$ and what does the Cauchy-Schwarz inequality give?