Let $X$ and $Y$ be normed vector spaces. Show that a linear map $T: X \rightarrow Y$ is continuous if and only if it is bounded.

Now let $X, Y, Z$ be Banach spaces. We say that a map $F: X \times Y \rightarrow Z$ is bilinear

$$\begin{aligned} &F(\alpha x+\beta y, z)=\alpha F(x, z)+\beta F(y, z), \text { for all scalars } \alpha, \beta \text { and } x, y \in X, z \in Y \\ &F(x, \alpha y+\beta z)=\alpha F(x, y)+\beta F(x, z), \text { for all scalars } \alpha, \beta \text { and } x \in X, y, z \in Y . \end{aligned}$$

Suppose that $F$ is bilinear and is continuous in each variable separately. Show that there exists a constant $M \geqslant 0$ such that

$$\|F(x, y)\| \leqslant M\|x\|\|y\|$$

for all $x \in X, y \in Y$.

[Hint: For each fixed $x \in X$, consider the map $y \mapsto F(x, y)$ from $Y$ to $Z$.]