Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 22I / a / Solution

The bounded inverse theorem states that a bounded bijective linear map between Banach spaces has a bounded inverse.

The closed graph theorem states that if $X$ and $Y$ are Banach spaces and $T:X\to Y$ is linear, then $T$ is continuous if and only if its graphis closed in the Banach space $X\times Y$.

If $T$ is continuous and $(x_n,T{x}_n)\to (x,y)$, then $T{x}_n\to Tx$, so uniqueness of limits gives $y=Tx$. Hence the graph is closed.

Conversely, suppose $\Gamma (T)$ is closed. It is then a Banach space. The coordinate projectionis bounded and bijective. By the bounded inverse theorem, $P_X^{-1}:x\mapsto (x,Tx)$ is bounded. Composing it with the bounded second-coordinate projection givesso $T$ is continuous.

Solved by gpt-5.6-sol high.