State and prove the principle of uniform boundedness.

[You may assume the Baire category theorem.]

Suppose that $X, Y$ and $Z$ are Banach spaces. Suppose that

$$F: X \times Y \rightarrow Z$$

is linear and continuous in each variable separately, that is to say that, if $y$ is fixed,

$$F(\cdot, y): X \rightarrow Z$$

is a continuous linear map and, if $x$ is fixed,

$$F(x, \cdot): Y \rightarrow Z$$

is a continuous linear map. Show that there exists an $M$ such that

$$\|F(x, y)\|_{Z} \leqslant M\|x\|_{X}\|y\|_{Y}$$

for all $x \in X, y \in Y$. Deduce that $F$ is continuous.

Suppose $X, Y, Z$ and $W$ are Banach spaces. Suppose that

$$G: X \times Y \times W \rightarrow Z$$

is linear and continuous in each variable separately. Does it follow that $G$ is continuous? Give reasons.

Suppose that $X, Y$ and $Z$ are Banach spaces. Suppose that

$$H: X \times Y \rightarrow Z$$

is continuous in each variable separately. Does it follow that $H$ is continuous? Give reasons.