Norm and Normed Spaces

In General > s.a. Banach; sequences [Cauchy]; Hölder Inequality.
$ Norm: A mapping  ·  : V → 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 (i)–(ii).
$ Normed space: A pair (V,  · ) as above.
* Equivalent norms: N₁ and N₂ are equivalent when there exist c and C in R such that for all v in V, cN₂(v) N₁(v) CN₂(v).

Relationships
* And topology: Any normed space can be given a topology by B_R,x:= {y in V | y–x < R}.
* And distance: A normed space can be given a distance by d(x, y):= x–y.

Examples > s.a. tensors [on tensor product spaces].
* On Rⁿ: Can define the L^p norm x_p:= [_i |xⁱ|^p]^1/p; as p → , we get x_infty:= max_i |xⁱ|; All of these norms are equivalent, and induce the Euclidean topology on Rⁿ.
* On operators on a Banach space X: For an operator A: X → X, A:= sup_{v in X} Av_X / v_X .
* On maps between classes of operators: For a completely positive map between Schatten p and q classes of operators, the p → q norm is := max_A A_q / A_p .
@ For operators: Prodan et al mp/05 [compact complex symmetric operator, computation]; Watrous QIC(05), Audenaert mp/05 [p → q norms of completely positive maps].

References
@ General: Day 73.
@ Related topics: Busch LMP(98)mp [base normed vector spaces].

Generalizations
* Seminorm: A map  ·  :V → R, where V is a vector space over C, such that x+y x + y, and x = || x; Positivity follows, but not definiteness.

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