Norm and Normed Spaces  

In General > s.a. Banach; sequences [Cauchy]; Hölder Inequality.
$ Norm: A mapping || · || : V → \(\mathbb R\), with V a vector space, such that (i) ||av|| = |a| ||v||; (ii) ||v+w|| < ||v|| + ||w|| (triangle inequality); (iii) ||v|| ≥ 0, for all v in V, and ||v|| = 0 implies v = 0; Positivity follows from conditions (i)–(ii).
$ Normed space: A pair (V, || · ||) as above.
* Equivalent norms: N1 and N2 are equivalent when there exist c and C in \(\mathbb R\) such that for all v in V, cN2(v) ≤ N1(v) ≤ CN2(v).
> Online resources: see Wikipedia page.

Relationships
* And topology: Any normed space can be given a ("strong") topology by defining the base of open balls BR,x:= {yV | ||yx|| < R}.
* And distance: A normed space can be given a distance by d(x, y):= ||xy||.

Examples > s.a. tensors [on tensor product spaces].
* On Rn: Can define the Lp norm ||x||p:= [∑i |xi| p]1/p; as p → ∞, we get ||x||:= maxi |xi|; All of these norms are equivalent, and induce the Euclidean topology on \(\mathbb R\)n.
* On function spaces: A common type of norms are the Sobolev norms; > s.a. functional analysis.
* On operators on a Banach space X: For an operator A: XX, ||A||:= supvX ||Av||X / ||v||X .
* On maps between classes of operators: For a completely positive map Φ between Schatten p and q classes of operators, the p q norm is ||Φ||:= maxA ||ΦA||q / ||A||p .
@ For operators: Prodan et al JPA(06)mp/05 [compact complex symmetric operator, computation]; Watrous QIC(05), Audenaert LAA-mp/05 [p q norms of completely positive maps].

References
@ General: Day 73; Alsina et al 09 [characterizations of real normed spaces as inner product spaces].
@ Related topics: Busch LMP(98)mp [base normed vector spaces].

Generalizations
* Seminorm: A map || · || :V → \(\mathbb R\), where V is a vector space over \(\mathbb C\), such that ||x+y|| ≤ ||x|| + ||y||, and ||αx|| = |α| ||x||; Positivity follows, but not definiteness; A family Γ of seminorms on V defines a unique topology TΓ compatible with the vector structure of V; The topology TΓ is the largest making all the seminorms continuous, but it is not necessarily Hausdorff.


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