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__: *N*_{1}
and *N*_{2} are equivalent when there
exist* c* and *C* in \(\mathbb R\) such that for all *v* in *V*,
*cN*_{2}(*v*) ≤ *N*_{1}(*v*)
≤ *CN*_{2}(*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 *B*_{R,x}:=
{*y* ∈ *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 R__

*

*

*

@

**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 *T*_{Γ} compatible with the vector structure of *V*; The topology *T*_{Γ} is the largest making all the seminorms continuous, but it is not necessarily Hausdorff.

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