Kit for def:normed-space

In the setting of Routine Facts, Continued: Typed Arithmetic, Maps and Predicativity, let $n\ge1$ be a metatheoretic number, let $x_{0},\dots,x_{n-1}$ be distinct set variables, and let $i,j,k,l<n$, not necessarily distinct. A condition on $x_{0},\dots,x_{n-1}$ is predicative if it is expressed by a predicative formula, with $x_{0},\dots,x_{n-1}$ in place of the set variables listed in Conditions Expressed by a Predicative Formula, and Guards (i) and no displayed variables; $\mathbb{R}$, $+$, $\cdot$, $0$, $1$ and $\le$ are parameters. Below, $V=x_{i}$, the addition of $V$ is $x_{j}$, its scalar multiplication is $x_{k}$, and $\lVert\cdot\rVert=x_{l}$; $\Gamma$ is the standing hypothesis of Norms and Real Normed Spaces: "$V$, with addition $x_{j}$ and scalar multiplication $x_{k}$, is a real vector space, and $\lVert\cdot\rVert$ is a function from $V$ to $\mathbb{R}$"; $0_{V}$ is the zero vector of $V$, and $|\lambda|$ the absolute value of $\lambda\in\mathbb{R}$.

(i) Nonnegativity. The condition "$\Gamma$, and $\lVert v\rVert\ge0$ for every $v\in V$" on $x_{0},\dots,x_{n-1}$, from Norms and Real Normed Spaces (i), is predicative.

(ii) Definiteness. The condition "$\Gamma$, and $v=0_{V}$ for every $v\in V$ with $\lVert v\rVert=0$" on $x_{0},\dots,x_{n-1}$, from Norms and Real Normed Spaces (ii), is predicative.

(iii) Absolute homogeneity. The condition "$\Gamma$, and $\lVert\lambda v\rVert=|\lambda|\,\lVert v\rVert$ for every $\lambda\in\mathbb{R}$ and every $v\in V$" on $x_{0},\dots,x_{n-1}$, from Norms and Real Normed Spaces (iii), is predicative.

(iv) Triangle inequality. The condition "$\Gamma$, and $\lVert u+v\rVert\le\lVert u\rVert+\lVert v\rVert$ for all $u,v\in V$" on $x_{0},\dots,x_{n-1}$, from Norms and Real Normed Spaces (iv), is predicative.

(v) Norms. The condition "$\Gamma$, and $\lVert\cdot\rVert$ is a norm on $V$" on $x_{0},\dots,x_{n-1}$ is predicative.

(vi) Real normed spaces. The condition "$\Gamma$, and $(V,\lVert\cdot\rVert)$, with addition $x_{j}$ and scalar multiplication $x_{k}$, is a real normed space" on $x_{0},\dots,x_{n-1}$ is predicative.