Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 2 / 23G (Analysis of Functions)

A functional $p:X\to \mathbb{R}$ is sublinear when

Any extension to $\tilde{M}=M+\mathbb{R}x$ must have the formChoose $c$ satisfyingThis interval is nonempty: for $y,z\in M$,which rearranges to the required lower bound being at most the upper bound. For $t>0$, positive homogeneity reduces
$\tilde{\ell }(y+tx)\le p(y+tx)$ to the upper inequality after replacing $y$ by $y/t$; for $t<0$ it reduces to the lower inequality. The case $t=0$ is the hypothesis on $\ell$. Thus this choice gives the required dominated linear extension.

The dominated form of the Hahn-Banach theorem states that a linear functional on a subspace, bounded above by a sublinear functional, extends linearly to the whole real vector space while retaining that bound.

Let $M=\mathrm{span}\{z_1,\dots ,z_n\}$. Its coordinate mapsare continuous because every linear map on a finite-dimensional normed space is continuous. Hahn--Banach extends each to
${\ell }_j\in Z^{\prime }$ without increasing its norm, and then

For an arbitrary finite-dimensional subspace $M$, choose a basis $z_1,\dots ,z_n$ and these extended coordinate functionals. Thenis closed. Every $z\in Z$ has the decompositionApplying every ${\ell }_j$ shows $M\cap N=\{0\}$, so