The Euclidean norm and its square are convex, with a ball of subgradients at zero for the norm

Given: The Euclidean norm The $p$-norms $\parallel x{\parallel }_p$ for rational $p\ge 1$, and $\parallel x{\parallel }_{\mathrm{\infty }}$, convexity Convex and strictly convex functions on Euclidean convex sets, and uniqueness of the subgradient of a differentiable convex function The subdifferential of a differentiable convex function is its gradient singleton.

The Euclidean norm is a norm on ${\mathbb{R}}^n$ and therefore satisfies the triangle inequality and homogeneity (Cauchy-Schwarz $\mid ⟨x,y⟩\mid \le \parallel x{\parallel }_2\parallel y{\parallel }_2$ with its equality case, the triangle inequality for $\parallel \cdot {\parallel }_2$, the parallelogram law and polarisation).

A $C^2$ function on an open convex set is convex if and only if its Hessian is positive semidefinite everywhere (A $C^2$ function is convex exactly when its Hessian is positive semidefinite).

A $C^2$ function on an open convex set whose Hessian is positive definite everywhere is strictly convex (An everywhere-positive-definite Hessian implies strict convexity).

The Euclidean inner product satisfies $\mathrm{\mid }⟨v,y⟩\mathrm{\mid }\le \mathrm{\parallel }v{\mathrm{\parallel }}_2\mathrm{\parallel }y{\mathrm{\parallel }}_2$ (Cauchy-Schwarz $\mid ⟨x,y⟩\mid \le \parallel x{\parallel }_2\parallel y{\parallel }_2$ with its equality case, the triangle inequality for $\parallel \cdot {\parallel }_2$, the parallelogram law and polarisation).

The derivative of $x^d$ is $d{x}^{d-1}$ for every integer $d\ge 1$ (For a natural $n\ge 1$ the function $x\mapsto x^n$ is differentiable everywhere with derivative $\iota (n)x^{n-1}$; for $n=0$ it is the constant $1$, with derivative $0$; for a natural $n\ge 1$ the function $x\mapsto x^{-n}$ is differentiable at every $x\mathrm{\ne }0$ with derivative $-\iota (n)x^{-n-1}$; consequently every polynomial function is differentiable at every real, with the derivative computed term by term).

Sums and scalar multiples of differentiable real functions are differentiable with the expected derivatives (Sums, scalar multiples, products and quotients: $(f+g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)+g^{\mathrm{\prime }}(c)$, $(\alpha f{)}^{\mathrm{\prime }}(c)=\alpha f^{\mathrm{\prime }}(c)$, $(fg{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)g(c)+f(c)g^{\mathrm{\prime }}(c)$, and $(f\mathrm{/}g{)}^{\mathrm{\prime }}(c)=(f^{\mathrm{\prime }}(c)g(c)-f(c)g^{\mathrm{\prime }}(c))\mathrm{/}g(c{)}^2$ when $g(c)\mathrm{\ne }0$).

Sums and scalar multiples of totally differentiable Euclidean maps are totally differentiable with the expected derivatives (Sums and scalar multiples of totally differentiable maps are totally differentiable with the expected derivatives).