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 by
‖
x
+
y
‖
≤
K
(
‖
x
‖
+
‖
y
‖
)
{\displaystyle \|x+y\|\leq K(\|x\|+\|y\|)}
for some
K
>
1.
{\displaystyle K>1.}
Definition
A quasi-seminorm on a vector space
X
{\displaystyle X}
is a real-valued map
p
{\displaystyle p}
on
X
{\displaystyle X}
that satisfies the following conditions:
Non-negativity :
p
≥
0
;
{\displaystyle p\geq 0;}
Absolute homogeneity :
p
(
s
x
)
=
|
s
|
p
(
x
)
{\displaystyle p(sx)=|s|p(x)}
for all
x
∈
X
{\displaystyle x\in X}
and all scalars
s
;
{\displaystyle s;}
there exists a real
k
≥
1
{\displaystyle k\geq 1}
such that
p
(
x
+
y
)
≤
k
p
(
x
)
+
p
(
y
)
{\displaystyle p(x+y)\leq k[p(x)+p(y)]}
for all
x
,
y
∈
X
.
{\displaystyle x,y\in X.}
If
k
=
1
{\displaystyle k=1}
then this inequality reduces to the
triangle inequality . It is in this sense that this condition generalizes the usual triangle inequality.
A quasinorm is a quasi-seminorm that also satisfies:
Positive definite /Point-separating : if
x
∈
X
{\displaystyle x\in X}
satisfies
p
(
x
)
=
0
,
{\displaystyle p(x)=0,}
then
x
=
0.
{\displaystyle x=0.}
A pair
(
X
,
p
)
{\displaystyle (X,p)}
consisting of a
vector space
X
{\displaystyle X}
and an associated quasi-seminorm
p
{\displaystyle p}
is called a quasi-seminormed vector space .
If the quasi-seminorm is a quasinorm then it is also called a quasinormed vector space .
Multiplier
The
infimum of all values of
k
{\displaystyle k}
that satisfy condition (3) is called the multiplier of
p
.
{\displaystyle p.}
The multiplier itself will also satisfy condition (3) and so it is the unique smallest real number that satisfies this condition.
The term
k
{\displaystyle k}
-quasi-seminorm is sometimes used to describe a quasi-seminorm whose multiplier is equal to
k
.
{\displaystyle k.}
A
norm (respectively, a
seminorm ) is just a quasinorm (respectively, a quasi-seminorm) whose multiplier is
1.
{\displaystyle 1.}
Thus every
seminorm is a quasi-seminorm and every
norm is a quasinorm (and a quasi-seminorm).
Topology
If
p
{\displaystyle p}
is a quasinorm on
X
{\displaystyle X}
then
p
{\displaystyle p}
induces a vector topology on
X
{\displaystyle X}
whose neighborhood basis at the origin is given by the sets:
{
x
∈
X
:
p
(
x
)
<
1
/
n
}
{\displaystyle \{x\in X:p(x)<1/n\}}
as
n
{\displaystyle n}
ranges over the positive integers.
A
topological vector space with such a topology is called a quasinormed topological vector space or just a quasinormed space .
Every quasinormed topological vector space is
pseudometrizable .
A
complete quasinormed space is called a quasi-Banach space . Every
Banach space is a quasi-Banach space, although not conversely.
Related definitions
A quasinormed space
(
A
,
‖
⋅
‖
)
{\displaystyle (A,\|\,\cdot \,\|)}
is called a quasinormed algebra if the vector space
A
{\displaystyle A}
is an
algebra and there is a constant
K
>
0
{\displaystyle K>0}
such that
‖
x
y
‖
≤
K
‖
x
‖
⋅
‖
y
‖
{\displaystyle \|xy\|\leq K\|x\|\cdot \|y\|}
for all
x
,
y
∈
A
.
{\displaystyle x,y\in A.}
A
complete quasinormed algebra is called a quasi-Banach algebra .
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.
L
p
{\displaystyle L^{p}}
spaces with
0
<
p
<
1
{\displaystyle 0<p<1}
The
L
p
{\displaystyle L^{p}}
spaces for
0
<
p
<
1
{\displaystyle 0<p<1}
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
0
<
p
<
1
,
{\displaystyle 0<p<1,}
the
Lebesgue space
L
p
(
0
,
1
)
{\displaystyle L^{p}([0,1])}
is a
complete
metrizable TVS (an
F-space ) that is not
locally convex (in fact, its only
convex open subsets are itself
L
p
(
0
,
1
)
{\displaystyle L^{p}([0,1])}
and the empty set) and the only
continuous linear functional on
L
p
(
0
,
1
)
{\displaystyle L^{p}([0,1])}
is the constant
0
{\displaystyle 0}
function (
Rudin 1991 , §1.47).
In particular, the
Hahn-Banach theorem does not hold for
L
p
(
0
,
1
)
{\displaystyle L^{p}([0,1])}
when
0
<
p
<
1.
{\displaystyle 0<p<1.}
See also
References
Aull, Charles E.; Robert Lowen (2001). Handbook of the History of General Topology .
Springer .
ISBN
0-7923-6970-X .
Conway, John B. (1990). A Course in Functional Analysis .
Springer .
ISBN
0-387-97245-5 .
Kalton, N. (1986).
"Plurisubharmonic functions on quasi-Banach spaces" (PDF) . Studia Mathematica . 84 (3). Institute of Mathematics, Polish Academy of Sciences: 297–324.
doi :
10.4064/sm-84-3-297-324 .
ISSN
0039-3223 .
Nikolʹskiĭ, Nikolaĭ Kapitonovich (1992). Functional Analysis I: Linear Functional Analysis . Encyclopaedia of Mathematical Sciences. Vol. 19.
Springer .
ISBN
3-540-50584-9 .
Rudin, Walter (1991).
Functional Analysis . International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY:
McGraw-Hill Science/Engineering/Math .
ISBN
978-0-07-054236-5 .
OCLC
21163277 .
Swartz, Charles (1992). An Introduction to Functional Analysis .
CRC Press .
ISBN
0-8247-8643-2 .
Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces . Mineola, New York: Dover Publications, Inc.
ISBN
978-0-486-49353-4 .
OCLC
849801114 .
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