Banach lattice

In mathematics, specifically in functional analysis and order theory, a Banach lattice $(X,\|\cdot \|)$ is a Riesz space with a norm $\|\cdot \|$ such that $(X,\|\cdot \|)$ is a Banach space and for all $x, y \in X$ the implication $|x|\leq |y|\Rightarrow \|x\|\leq \|y\|$ holds, where as usual $|x|:=x\vee -x$ .

Examples and constructions

$\mathbb {R}$ , together with its absolute value as a norm, is a Banach lattice.
Let $X$ be a topological space, $Y$ a Banach lattice and ${\mathcal {C}}(X,Y)$ the space of bounded, continuous functions from $X$ to $Y$ with norm $\|f\|_{\infty }:=\sup _{x\in X}\|f(x)\|_{Y}$ . ${\mathcal {C}}(X,Y)$ becomes a Banach lattice with the pointwise order $f\leq g:\Leftrightarrow \forall x\in X:f(x)\leq g(x)$ .

References

Abramovich, Yuri A.; Aliprantis, C. D. (2002). An Invitation to Operator Theory. Graduate Studies in Mathematics. 50. American Mathematical Society. ISBN 0-8218-2146-6.

Functional analysis (topics)
TVS types	Banach Banach Lattice Barrelled Bornological Brauner F-space Finite-dimensional Fréchet (tame) Hilbert (pre-Hilbert Polarization identity) LF-space Locally convex (Seminorms/Minkowski functionals) Mackey Montel Nuclear Normed (norm) Quasinormed Reflexive Riesz Smith Stereotype Strictly convex Webbed Topological tensor product (of Hilbert spaces)
Mapping topologies	Dual Dual space (Dual norm) Operator Ultraweak Weak (polar operator) Mackey Strong (polar operator) Ultrastrong Uniform convergence
Linear operators	Adjoint Bilinear (form operator sesquilinear) (Un)Bounded Closed Compact (Dis)Continuous Densely defined Fredholm Hilbert–Schmidt Functionals (positive) Normal Nuclear Self-adjoint Strictly singular Trace class Transpose Unitary
Operator theory	Banach algebras C-algebras Spectrum (C-algebra radius) Spectral theory (of ODEs Spectral theorem) Polar decomposition Singular value decomposition
Theorems	Banach–Alaoglu Banach–Mazur Banach–Saks Bessel's inequality Cauchy–Schwarz inequality Closed graph Closed range Eberlein–Šmulian Freudenthal spectral Gelfand–Mazur Gelfand–Naimark Goldstine Hahn–Banach (hyperplane separation) Kakutani fixed-point Krein–Milman Lomonosov's invariant subspace Mackey–Arens Mazur's lemma M. Riesz extension Riesz representation Open mapping Parseval's identity Schauder fixed-point
Analysis	Abstract Wiener space Bochner space Differentiation in Fréchet spaces Derivatives (Fréchet Gâteaux functional holomorphic) Integrals (Bochner Dunford Gelfand–Pettis regulated Paley–Wiener weak) Functional calculus (Borel continuous holomorphic) Inverse function theorem (Nash–Moser theorem) Measures (Lebesgue Projection-valued Vector) Weakly measurable function
Types of sets	Absolutely convex Absorbing Balanced Bounded Convex Convex cone (subset) Linear cone (subset) Radial Star-shaped Symmetric Zonotope
Subsets / set operations	Algebraic interior (core) Bounding points Convex hull Extreme point Interior Minkowski addition Polar

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Banach lattice

Examples and constructions

See also

References