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