Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 22I / a / Solution

$X^{\ast }$ is the Banach space of bounded linear functionals. For $T:X\to Y$, define $T^{\ast }:Y^{\ast }\to X^{\ast }$ by $T^{\ast }f=f\circ T$. It is linear and $\Vert T^{\ast }f\Vert \le \Vert T\Vert \Vert f\Vert$, so bounded (indeed $\Vert T^{\ast }\Vert =\Vert T\Vert$).

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