Norm and Normed Vector Spaces| 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:
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:=
{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 Rn:
One can define the Lp
norm ||x||p:=
[∑i
\(|x_i|^p\)]1/p; as p
→ ∞, we get ||x||∞:=
max\(_i\) |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: X →
X, ||A||:= supv
∈ X ||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
> 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 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.
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 13 apr 2019