Quasinorm Explained
In linear algebra, functional analysis and related areas of mathematics, a quasinorm is similar to a norm in that it satisfies the norm axioms, except that the triangle inequality is replaced byfor some
Definition
A on a vector space
is a real-valued map
on
that satisfies the following conditions:
- :
- :
for all
and all scalars
- there exists a real
such that
for all
then this inequality reduces to the triangle inequality. It is in this sense that this condition generalizes the usual triangle inequality.
A is a quasi-seminorm that also satisfies:
- Positive definite/: if
satisfies
then
A pair
consisting of a
vector space
and an associated quasi-seminorm
is called a . If the quasi-seminorm is a quasinorm then it is also called a .
Multiplier
The infimum of all values of
that satisfy condition (3) is called the of
The multiplier itself will also satisfy condition (3) and so it is the unique smallest real number that satisfies this condition. The term is sometimes used to describe a quasi-seminorm whose multiplier is equal to
A (respectively, a) is just a quasinorm (respectively, a quasi-seminorm) whose multiplier is
Thus every
seminorm is a quasi-seminorm and every
norm is a quasinorm (and a quasi-seminorm).
Topology
If
is a quasinorm on
then
induces a vector topology on
whose neighborhood basis at the origin is given by the sets:
as
ranges over the positive integers. A
topological vector space with such a topology is called a or just a .
Every quasinormed topological vector space is pseudometrizable.
A complete quasinormed space is called a . Every Banach space is a quasi-Banach space, although not conversely.
Related definitions
See also: Banach algebra.
A quasinormed space
is called a if the vector space
is an
algebra and there is a constant
such that
for all
A complete quasinormed algebra is called a .
Characterizations
A topological vector space (TVS) is a quasinormed space if and only if it has a bounded neighborhood of the origin.
Examples
Since every norm is a quasinorm, every normed space is also a quasinormed space.
spaces with
The
spaces for
are quasinormed spaces (indeed, they are even
F-spaces) but they are not, in general, normable (meaning that there might not exist any norm that defines their topology). For
the
Lebesgue space
is a
complete metrizable TVS (an
F-space) that is
locally convex (in fact, its only
convex open subsets are itself
and the empty set) and the continuous linear functional on
is the constant
function . In particular, the
Hahn-Banach theorem does hold for
when
References
- Book: Aull, Charles E.. Robert Lowen. Handbook of the History of General Topology. 2001. 0-7923-6970-X. Springer.
- Book: Conway, John B.. A Course in Functional Analysis. 0-387-97245-5. Springer. 1990.
- Kalton. N.. Plurisubharmonic functions on quasi-Banach spaces. Studia Mathematica. Institute of Mathematics, Polish Academy of Sciences. 84. 3. 1986. 0039-3223. 10.4064/sm-84-3-297-324. 297–324.
- Book: Nikolʹskiĭ, Nikolaĭ Kapitonovich. Functional Analysis I: Linear Functional Analysis. 3-540-50584-9. 1992. Springer. Encyclopaedia of Mathematical Sciences. 19.
- Book: Swartz, Charles. An Introduction to Functional Analysis. 1992. CRC Press. 0-8247-8643-2.