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- general topology - Proof of the Banach–Alaoglu theorem - Mathematics . . .
The Banach–Alaoglu theorem states that the closed unit ball of $B'$ (where $B'$ is the dual to a Banach space $B$ over a field) is compact in the weak* topology
- Banach limit, Hahn-Banach theorem - Mathematics Stack Exchange
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- Prove that the Set of Bounded Linear Operators is Banach
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- Banach Indicatrix Function - Mathematics Stack Exchange
The argument I present is Banach's, as given in Sur les lignes rectifiables et les surfaces dont l'aire est finie, Fund Math 7 (1925), 225–236 (Théorèmes 1 and 2), which I reproduce here for the convenience of the readers who don't read French
- Infinite-dimensional manifolds: Fréchet, Banach and Hilbert manifolds . . .
On Banach-Lie groups acting on finite dimensional manifolds Tohoku Mathematical Journal, Second Series, 30(2), 223-250 ] that states: if G is a Banach Lie group acting faithfully and transitively on a finite dimensional manifold M, then G has to be finite dimensional!
- What is the relation between a Banach space and a Hilbert space?
$\begingroup$ This is a bit vague, but there are useful remarks that could be made in answers to it Specifically, Hilbert spaces have a (true!) minimum Dirichlet principle, and Banach spaces easily and non-pathologically fail this (e g , the literally incorrect Dirichlet principle that was very important throughout the late 19th century, and only reformulated in 1905 by Giuseppe ("Beppo
- Showing that space of absolutely continuous functions is Banach space
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- functional analysis - Why isometric isomorphic between Banach spaces . . .
Each mathematical theory study their own objects and maps between them In our case these are Banach spaces and bounded linear maps We choose linear maps because we want them to preserve linear structure of Banach spaces We choose them bounded to nicely interact with the norm structure of Banach spaces
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