Proof of Elementary Properties of Sublinear Functionals

For (i): $0_{V}=0\cdot0_{V}$ by Elementary Identities in a Vector Space (ii), so Sublinear Functional (ii) with $\lambda=0$ gives

$$p(0_{V})=p(0\cdot0_{V})=0\cdot p(0_{V})=0 .$$

For (ii), let $x\in V$. Since $x+(-x)=0_{V}$ (The Zero Vector, Negatives and Differences in a Vector Space (ii)), the value at zero and Sublinear Functional (i) give

$$0=p(0_{V})=p\bigl(x+(-x)\bigr)\le p(x)+p(-x),$$

that is, $-p(-x)\le p(x)$.

For (iii), let $M$ and $f$ be as there and let $x\in M$. The upper inequality $f(x)\le p(x)$ is the hypothesis. For the lower one, $-x=(-1)x$ by Elementary Identities in a Vector Space (iv), so $-x\in M$ by Linear Subspace of a Real Vector Space (iii), and $f(-x)=-f(x)$ by Linear Functional on a Linear Subspace (ii). The hypothesis at $-x$ gives $-f(x)=f(-x)\le p(-x)$, that is, $-p(-x)\le f(x)$.

For (iv), let $\lVert\cdot\rVert$ be a norm on $V$. It is subadditive by Norms and Real Normed Spaces (iv), and for every real $\lambda\ge0$ and $x\in V$, Norms and Real Normed Spaces (iii) gives $\lVert\lambda x\rVert=|\lambda|\,\lVert x\rVert=\lambda\lVert x\rVert$. So $\lVert\cdot\rVert$ is a sublinear functional.

For (v), let $\lVert\cdot\rVert$ and $C$ be as there and let $x,y\in V$. Since $x=y+(x-y)$ (Elementary Identities in a Vector Space (viii)), Sublinear Functional (i) and the hypothesis give

$$p(x)\le p(y)+p(x-y)\le p(y)+C\lVert x-y\rVert .$$

Exchanging the roles of $x$ and $y$ gives $p(y)\le p(x)+C\lVert y-x\rVert$, and $\lVert y-x\rVert=\lVert x-y\rVert$: indeed $y-x=-(x-y)=(-1)(x-y)$ by Elementary Identities in a Vector Space (xi) and Elementary Identities in a Vector Space (iv), and Norms and Real Normed Spaces (iii) applies with $\lambda=-1$. The two inequalities together give $|p(x)-p(y)|\le C\lVert x-y\rVert$.

For (vi), let $\lVert\cdot\rVert$, $C$, $M$ and $f$ be as there and let $x\in M$. By (iii) and the hypothesis on $p$,

$$f(x)\le p(x)\le C\lVert x\rVert,\qquad -f(x)\le p(-x)\le C\lVert -x\rVert=C\lVert x\rVert,$$

where $\lVert -x\rVert=\lVert(-1)x\rVert=\lVert x\rVert$ by Elementary Identities in a Vector Space (iv) and Norms and Real Normed Spaces (iii). Hence $|f(x)|\le C\lVert x\rVert$.