List of Banach spaces

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 Diestel (1984, Chapter VII), the classical Banach spaces are those defined by Dunford & Schwartz (1958), which is the source for the following table.

Here K denotes the field of real numbers or complex numbers and I is a closed and bounded interval [a,b]. The number p is a real number with 1 < p < ∞, and q is its Hölder conjugate (also with 1 < q < ∞), so that the next equation holds:

{\frac {1}{q}}+{\frac {1}{p}}=1,

and thus

q=\frac{p}{p-1} .

The symbol Σ denotes a σ-algebra of sets, and Ξ denotes just an algebra of sets (for spaces only requiring finite additivity, such as the ba space). The symbol μ denotes a positive measure: that is, a real-valued positive set function defined on a σ-algebra which is countably additive.

	Dual space	Reflexive	weakly complete	Norm	Notes
Classical Banach spaces
Kⁿ	Kⁿ	Yes	Yes	$\\|x\\|_2 = \left(\sum_{i=1}^n \|x_i\|^2\right)^{1/2}$
ℓⁿ_p	ℓⁿ_q	Yes	Yes	$\\|x\\|_p = \left(\sum_{i=1}^n \|x_i\|^p\right)^{1/p}$
ℓⁿ_∞	ℓⁿ₁	Yes	Yes	$\\|x\\|_\infty = \max_{1\le i\le n} \|x_i\|$
ℓ_p	ℓ_q	Yes	Yes	$\\|x\\|_p = \left(\sum_{i=1}^\infty \|x_i\|^p\right)^{1/p}$	1 < p < ∞
ℓ₁	ℓ_∞	No	Yes	$\\|x\\|_{1}=\sum _{i=1}^{\infty }\|x_{i}\|$
ℓ_∞	ba	No	No	$\\|x\\|_\infty = \sup_i \|x_i\|$
c	ℓ₁	No	No	$\\|x\\|_\infty = \sup_i \|x_i\|$
c₀	ℓ₁	No	No	$\\|x\\|_\infty = \sup_i \|x_i\|$	Isomorphic but not isometric to c.
bv	ℓ₁ + K	No	Yes	$\\|x\\|_{bv}=\|x_{1}\|+\sum _{i=1}^{\infty }\|x_{i+1}-x_{i}\|$
bv₀	ℓ₁	No	Yes	$\\|x\\|_{bv_{0}}=\sum _{i=1}^{\infty }\|x_{i+1}-x_{i}\|$
bs	ba	No	No	$\\|x\\|_{bs} = \sup_n\left\|\sum_{i=1}^nx_i\right\|$	Isometrically isomorphic to ℓ_∞.
cs	ℓ₁	No	No	$\\|x\\|_{bs} = \sup_n\left\|\sum_{i=1}^nx_i\right\|$	Isometrically isomorphic to c.
B(X, Ξ)	ba(Ξ)	No	No	$\\|f\\|_B = \sup_{x\in X}\|f(x)\|$
C(X)	rca(X)	No	No	$\\|f\\|_{B} = \sup_{x\in X}\left\|f(x)\right\|$	X is a compact Hausdorff space.
ba(Ξ)	?	No	Yes	$\\|\mu\\|_{ba} = \sup_{A\in\Sigma} \|\mu\|(A)$ (variation of a measure)
ca(Σ)	?	No	Yes	$\\|\mu\\|_{ba} = \sup_{A\in\Sigma} \|\mu\|(A)$
rca(Σ)	?	No	Yes	$\\|\mu\\|_{ba} = \sup_{A\in\Sigma} \|\mu\|(A)$
L^p(μ)	L^q(μ)	Yes	Yes	$\\|f\\|_p = \left\{\int \|f\|^p\,d\mu\right\}^{1/p}$	1 < p < ∞
L¹(μ)	L^∞(μ)	No	?	$\\|f\\|_{1}=\int \|f\|\,d\mu$	If the measure μ on S is sigma-finite
L^∞(μ)	$N_\mu^\perp$	No	?	$\\|f\\|_\infty \equiv \inf \{ C\ge 0 : \|f(x)\| \le C \mbox{ for almost every } x\}.$	where $N_\mu^\perp =\{\sigma\in ba(\Sigma) : \lambda \ll \mu\}$
BV(I)	?	No	Yes	$\\|f\\|_{BV} = \lim_{x\to a^+}f(x) + V_f(I)$	V_f(I) is the total variation of f.
NBV(I)	?	No	Yes	$\\|f\\|_{BV}=V_{f}(I)$	NBV(I) consists of BV functions such that $\lim_{x\to a^+}f(x)=0$ .
AC(I)	K+L^∞(I)	No	Yes	$\\|f\\|_{BV} = \lim_{x\to a^+}f(x) + V_f(I)$	Isomorphic to the Sobolev space W^1,1(I).
Cⁿ[a,b]	rca([a,b])	No	No	$\\|f\\| = \sum_{i=0}^n \sup_{x\in [a,b]} \|f^{(i)}(x)\|.$	Isomorphic to Rⁿ ⊕ C([a,b]), essentially by Taylor's theorem.

Banach spaces in other areas of analysis

The Asplund spaces
The Hardy spaces
The space BMO of functions of bounded mean oscillation
The space of functions of bounded variation
Sobolev spaces
The Birnbaum–Orlicz spaces L_A(μ).
Hölder spaces C^k,α(Ω).
Lorentz 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 ℓ^p nor c₀ can be embedded.
W.T. Gowers construction of a space X that is isomorphic to $X\oplus X\oplus X$ but not $X\oplus X$ serves as a counterexample for weakening the premises of the Schroeder–Bernstein theorem ^[1]

Notes

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

References

Diestel, Joseph (1984), Sequences and series in Banach spaces, Springer-Verlag, ISBN 0-387-90859-5 .
Dunford, N.; Schwartz, J.T. (1958), Linear operators, Part I, Wiley-Interscience .

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.