Пространство ограниченных последовательностей

Пространство ограниченных последовательностей — метрическое пространство. Каждый его элемент определяется как бесконечная последовательность чисел $x=\{x_{n}\}_{i=1}^{\infty }$ , каждый член которой ограничен по модулю: $|x_{i}|\leqslant K_{i}$ , $i=1,...\infty$ , где $K_{i}$ , $i=1,...\infty$ - константы^[1], и в котором определено расстояние $\rho (x,y)$ между любыми двумя точками $x,y$ , как^[2]: $\rho (x,y)=\sup _{i}|x_{i}-y_{i}|$ , $i=1,...\infty$ , где $\sup$ - точная верхняя граница.

Для пространства ограниченных последовательностей приняты стандартные обозначения $m$ или $\ell _{\infty }$ ^[1].

Пространство $m$ не является сепарабельным^[3] и является полным^[4].

При определении нормы в $m$ как^[1]:

\left\|x\right\|=\sup _{i}|x_{i}|

i=1,...\infty

оно становится линейным нормированным пространством.

Примеры:

бесконечные последовательности чисел вида $x=\{x_{n}\}_{i=1}^{\infty }$ , таких, что $|x_{i}|\leqslant 1$ , $i=1,...\infty$
бесконечные последовательности чисел вида $x=\{x_{n}\}_{i=1}^{\infty }$ , таких, что $|x_{i}|\leqslant {\frac {1}{i}}$ , $i=1,...\infty$

См. также

Пространство квадратично-суммируемых последовательностей

Примечания

↑ ¹ ² ³ Крейн, 1972, с. 27.
↑ Соболев, 1968, с. 32.
↑ Соболев, 1968, с. 44.
↑ Соболев, 1968, с. 50.

Литература

Соболев В. И. Лекции по дополнительным главам математического анализа. — М.: Наука, 1968. — 288 с. — 70 000 экз.
Крейн С. Г. Функциональный анализ. — М.: Наука, 1972. — 544 с. — 29 000 экз.

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