S

Examples

Thereafter

\ mathbb {K}

can be replaced by

\ mathbb {R}

\ mathbb {C}

the Euclidean spaces $\ mathbb {K} ^n$ provides with the standard $||X|| = (\ sum_i (x_i) ^2) ^ {1/2} \,$ where $X = (x_1,…, x_n)$ is spaces of Banach.
the space of the continuous functions definite on an interval: $f: \ rightarrow \ mathbb {K}$ provided with the standard $||F|| = \ sup_ {X \ in} (|F (X)|)$ form a space of Banach.

Property of closed encased

Either a decreasing succession of closed not vacuums of a space of Banach such as the diameter of each closed or real and that the continuation of the diameters tends towards 0. Then the intersection of closed is reduced to a Singleton.

This property makes it possible to show that a space of Banach is of Baire.

To note that this property can be false without the assumption that the diameters tend towards 0, even if one supposes closed limited.

Theorem of Banach-Steinhaus

See the Leitartikel: Theorem of Banach-Steinhaus .

E a space of Banach, and $F$ are $a vector Space normalized. Either (u_i) _ {I \ in I} a family of elements of \ mathcal L (E, F) (see Linear application) and or has the whole of the vectors X \ in E such as \ sup_ {I \ in I} \|u_i (X) \| < + \ infty . Then is has is thin, i.e. countable meeting of rare units (a unit is rare if the Intérieur of sound adherence is empty) and its complementary is dense, that is to say \ sup_ {I \ in I} \| u_i \| < + \ infty . In particular, if A=E, only the second possibility is possible.$

Note: the last standard used is the standard of operator (or normalizes subordinate).

Literature

Stefan Banach : Theory of the linear operations. -- Warszawa 1932. (Monografie Matematyczne; 1) Zbl 0005.20901

Internal bonds

Topological structures:

Theorems of analysis:

Biography:

Stefan Banach

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