Every norm is convex, and the operator norm is a supremum of linear functions

ProofBeginner⏱ 15 minOptimizationconvexitynormssupremum

Problem

Prove that any norm is a convex function.
Prove that $A\mapsto\|A\|$ (spectral norm) is convex on $\R^{m\times n}$ , by exhibiting it as a supremum of linear functions of $A$ .
Deduce that $\lambda_{\max}$ is convex on symmetric matrices and $\lambda_{\min}$ is concave.

Write a complete argument. Every inequality you use should be one you could state precisely.

Opening this before you have a real attempt on paper is the fastest way to feel like you understand something you cannot reproduce.

Problem	Field	Type	Difficulty	Time
Convexity of the logistic loss	Classical ML	Proof	Beginner	15 min
Equality in Jensen's inequality	Foundations	Proof	Beginner	15 min
Existence and uniqueness of the minimiser	Optimization	Proof	Beginner	15 min
First-order characterisation of convexity	Optimization	Proof	Beginner	30 min
Lipschitz constants depend on the norm	Foundations	Numerical	Beginner	15 min