Norm and Normed Vector 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**
> s.a. modified quantum theory [with indefinite norm].

* __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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send feedback and suggestions to bombelli at olemiss.edu – modified 13 apr 2019