# Symbols:Analysis

## Symbols used in Analysis

### Norm

$\norm z$

Let $\struct {R, +, \circ}$ be a division ring with norm $\norm {\,\cdot\,}_R$.

Let $V$ be a vector space over $R$, with zero $0_V$.

A norm on $V$ is a map from $V$ to the nonnegative reals:

$\norm{\,\cdot\,}: V \to \R_{\ge 0}$

satisfying the (vector space) norm axioms:

 $(\text N 1)$ $:$ Positive Definiteness: $\ds \forall x \in V:$ $\ds \norm x = 0$ $\ds \iff$ $\ds x = \mathbf 0_V$ $(\text N 2)$ $:$ Positive Homogeneity: $\ds \forall x \in V, \lambda \in R:$ $\ds \norm {\lambda x}$ $\ds =$ $\ds \norm {\lambda}_R \times \norm x$ $(\text N 3)$ $:$ Triangle Inequality: $\ds \forall x, y \in V:$ $\ds \norm {x + y}$ $\ds \le$ $\ds \norm x + \norm y$

The $\LaTeX$ code for $\norm z$ is \norm z .