Бесконечномерное пространство

Бесконечномерное пространство — векторное пространство c бесконечно большой размерностью. Изучение бесконечномерных пространств и их отображений является главной задачей функционального анализа. Наиболее простыми бесконечномерными пространствами являются гильбертовы пространства, наиболее близкие по свойствам к конечномерным евклидовым пространствам^[1].

Определение

Линейное векторное пространство называется бесконечномерным, если для любого целого числа $N>0$ в нем найдется линейно независимая система, состоящая из $N$ векторов^[2]^[3].

Базис

Для бесконечномерного пространства существуют различные определения базиса. Так, например, базис Гамеля определяется, как множество векторов в линейном пространстве, таких, что любой вектор пространства может быть представлен в виде некоторой их конечной линейной комбинации единственным образом.

Для топологических векторных пространств можно определить базис Шаудера. Система элементов $\left\{e_{k}\right\}$ образует базис Шаудера пространства $E$ , если каждый элемент $x\in E$ представим единственным образом в виде сходящегося ряда $x=\sum _{k=1}^{\infty }c_{k}e_{k}$ ^[4]. Базис Шаудера существует не всегда.

Примеры

Линейное пространство непрерывных на данном промежутке функций^[2].
Гильбертово пространство, образованное бесконечной последовательностью чисел $x=(\xi _{1},...,\xi _{k},...)$ со сходящейся суммой квадратов $\sum _{n=1}^{\infty }\xi _{n}^{2}<\infty$ ^[5].
Множество всех многочленов (над данным полем)^[6].
Пространство квадратично-суммируемых последовательностей

Свойства

Бесконечномерное пространство не изоморфно никакому конечномерному^[7].

См. также

Конечномерное пространство

Примечания

↑ Функциональный анализ // Математический энциклопедический словарь / гл. ред. Ю. В. Прохоров. — М., Советская энциклопедия, 1988. — с. 613-615
↑ ¹ ² Ефимов, 2004, с. 33.
↑ Шикин Е. В. Линейные пространства и отображения. — М., МГУ, 1987. — с. 17
↑ Крейн, 1964, с. 74.
↑ Шилов, 1961, с. 182.
↑ Ефимов, 2004, с. 32.
↑ Ефимов, 2004, с. 39.

Литература

Ефимов Н.В., Розендорн Э.Р. Линейная алгебра и многомерная геометрия. — М.: Физматлит, 2004. — 464 с. — ISBN 5-9221-0386-5.
Шилов Г.Е. Математический анализ. Специальный курс. — М.: Наука, 1961. — 436 с.
под ред. Крейн С.Г. Функциональный анализ. — М.: Наука, 1964. — 424 с. — 17 500 экз.

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