List of Banach spaces explained

In the mathematical field of functional analysis, Banach spaces are among the most important objects of study. In other areas of mathematical analysis, most spaces which arise in practice turn out to be Banach spaces as well.

Classical Banach spaces

According to, the classical Banach spaces are those defined by, which is the source for the following table.

Banach spaces in other areas of analysis

• The Asplund spaces
• The Hardy spaces
• The space

\operatorname{BMO} of functions of bounded mean oscillation

• The space of functions of bounded variation
• Sobolev spaces
• The Birnbaum–Orlicz spaces

L ^{A(\mu).}

• Hölder spaces

C ^{k(\Omega).}

• Lorentz space
• ba space

Banach spaces serving as counterexamples

• James' space, a Banach space that has a Schauder basis, but has no unconditional Schauder Basis. Also, James' space is isometrically isomorphic to its double dual, but fails to be reflexive.
• Tsirelson space, a reflexive Banach space in which neither \ell ^{p} nor c _{0} can be embedded.
• W.T. Gowers construction of a space

X that is isomorphic to X ⊕ X ⊕ X but not X ⊕ X serves as a counterexample for weakening the premises of the Schroeder–Bernstein theorem^[1]

References

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Notes and References

1. W.T. Gowers, "A solution to the Schroeder–Bernstein problem for Banach spaces", Bulletin of the London Mathematical Society, 28 (1996) pp. 297–304.