linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. ...

functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...

and related areas of

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a quasinorm is similar to a

norm Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envir ...

in that it satisfies the norm axioms, except that the

triangle inequality In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of degenerate triangles, but ...

is replaced by

\, x + y\,  \leq K(\, x\,  + \, y\, )

for some

K > 0.

Related concepts

:Definition: A quasinorm on a vector space

X

is a real-valued map

p

X

that satisfies the following conditions:

Non-negativity: $p \geq 0;$
Absolute homogeneity In mathematics, a homogeneous function is a function of several variables such that, if all its arguments are multiplied by a scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneity, or simply the ''d ...
: $p(s x) = , s, p(x)$ for all $x \in X$ and all scalars $s;$
there exists a $k \geq 1$ such that $p(x + y) \leq k (x) + p(y) /math> for all x, y \in X.$

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:

\

n

ranges over the positive integers. A

topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...

(TVS) with such a topology is called a quasinormed space. Every quasinormed TVS is a pseudometrizable. A

vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...

with an associated quasinorm is called a quasinormed vector space. A

complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...

quasinormed space is called a quasi-Banach space. A quasinormed space

(A, \,  \,\cdot\, \, )

is called a quasinormed algebra if the vector space

A

is an

algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...

and there is a constant

K > 0

such that

\, x y\,  \leq K \, x\,  \cdot \, y\,

for all

x, y \in A.

A complete quasinormed algebra is called a quasi-Banach algebra.

Characterizations

(TVS) is a quasinormed space if and only if it has a bounded neighborhood of the origin.

References

* * * * * {{Topological vector spaces Linear algebra Norms (mathematics)

Related concepts

Characterizations

See also

References