فضاء متجهي معياري

Hierarchy of mathematical spaces. Normed vector spaces are a subset of metric spaces and a superset of inner product spaces.

الفضاء المتجهي المعياري Normed vector space هو فضاء اتجاهي عُرفت عليه دالة المعيار.

كل فضاء معياري هو فضاء متري ولكن العكس قد لا يتحقق.

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 تعريف

الفضاء المتجهي المعياري هو زوج (V، ‖·‖ ) حيث V هو فضاء متجهي و ‖·‖ معيار على V.

الفضاء المتجهي نصف المعياري هو زوج (p ،V) حيث V هو فضاء متجهي و p نصف معيار على V.

$\|x-y\|\geq |\|x\|-\|y\||$ مهما كانت المتجهتين x و y.

هذا يدل على أن دالة المعيار هي دالة متصلة.

 الفضاءات المعيارية كفضاءات قسمة للفراغات نصف المعيارية

The definition of many normed spaces (in particular, Banach spaces) involves a seminorm defined on a vector space and then the normed space is defined as the quotient space by the subspace of elements of seminorm zero. For instance, with the L^p spaces, the function defined by

$\|f\|_{p}=\left(\int |f(x)|^{p}\;dx\right)^{1/p}$

is a seminorm on the vector space of all functions on which the Lebesgue integral on the right hand side is defined and finite. However, the seminorm is equal to zero for any function supported on a set of Lebesgue measure zero. These functions form a subspace which we "quotient out", making them equivalent to the zero function.

 فضاءات الضرب المحدودة

Given n seminormed spaces X_i with seminorms q_i we can define the product space as

$X:=\prod _{i=1}^{n}X_{i}$

with vector addition defined as

$(x_{1},\ldots ,x_{n})+(y_{1},\ldots ,y_{n}):=(x_{1}+y_{1},\ldots ,x_{n}+y_{n})$

and scalar multiplication defined as

$\alpha (x_{1},\ldots ,x_{n}):=(\alpha x_{1},\ldots ,\alpha x_{n})$.

We define a new function q

$q:X\to \mathbb {R}$

for example as

$q:(x_{1},\ldots ,x_{n})\mapsto \sum _{i=1}^{n}q_{i}(x_{i})$.

which is a seminorm on X. The function q is a norm if and only if all q_i are norms.

More generally, for each real p≥1 we have the seminorm:

$q:(x_{1},\ldots ,x_{n})\mapsto \left(\sum _{i=1}^{n}q_{i}(x_{i})^{p}\right)^{\frac {1}{p}}.$

For each p this defines the same topological space.

A straightforward argument involving elementary linear algebra shows that the only finite-dimensional seminormed spaces are those arising as the product space of a normed space and a space with trivial seminorm. Consequently, many of the more interesting examples and applications of seminormed spaces occur for infinite-dimensional vector spaces.

 انظر أيضا

• فضاء باناخ
• فضاء الجداء الداخلي
• فضاء رياضي

 مراجع

• Rolewicz, Stefan (1987), Functional analysis and control theory: Linear systems, Mathematics and its Applications (East European Series), 29 (Translated from the Polish by Ewa Bednarczuk ed.), Dordrecht; Warsaw: D. Reidel Publishing Co.; PWN—Polish Scientific Publishers, pp. xvi+524, doi:10.1007/978-94-015-7758-8, ISBN 90-277-2186-6, OCLC 13064804