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 by$$\backslash |x\; +\; y\backslash |\; \backslash leq\; K(\backslash |x\backslash |\; +\; \backslash |y\backslash |)$$for some

K > 1 .

Definition

A on a vector space

X is a real-valued map p on X that satisfies the following conditions:

1. : p \geq 0 ;
2. : p(s x) = |s| p(x) for all x \in X and all scalars s;
3. there exists a real k \geq 1 such that p(x + y) \leq k [p(x) + p(y)] for all x, y \in X.
   • If
   k = 1 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:

4. Positive definite/: if
   x \in X satisfies p(x) = 0 , then x = 0 .

A pair

(X, p) consisting of a vector space X and an associated quasi-seminorm p is called a . If the quasi-seminorm is a quasinorm then it is also called a .

Multiplier

The infimum of all values of

k that satisfy condition (3) is called the of p. 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 k.

A (respectively, a) is just a quasinorm (respectively, a quasi-seminorm) whose multiplier is

1 . Thus every seminorm is a quasi-seminorm and every norm is a quasinorm (and a quasi-seminorm).

Topology

If

p is a quasinorm on X then p induces a vector topology on X whose neighborhood basis at the origin is given by the sets:$$\backslash$$as n 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

(A, \| ⋅ \|) is called a if the vector space A is an algebra and there is a constant K > 0 such that $$\backslash |x\; y\backslash |\; \backslash leq\; K\; \backslash |x\backslash |\; \backslash cdot\; \backslash |y\backslash |$$for all x, y \in A.

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.

L ^{p} spaces with 0 < p < 1

The

L ^{p} spaces for

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 , the Lebesgue space L ^{p([0,} 1 ]) is a complete metrizable TVS (an F-space) that is locally convex (in fact, its only convex open subsets are itself L ^{p([0,} 1 ]) and the empty set) and the continuous linear functional on L ^{p([0,} 1 ]) is the constant 0 function . In particular, the Hahn-Banach theorem does hold for L ^{p([0,} 1 ]) when 0 < p < 1 .

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.