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 $f\leq g:\Leftrightarrow \forall x\in X:f(x)\leq g(x)$ .

Functional analysis (topics)
TVS types	Banach Banach Lattice Barrelled Bornological Brauner F-space Finite-dimensional Fréchet (tame) Hilbert (pre-Hilbert) LF-space Locally convex Mackey Montel Nuclear Normed (norm) Quasinormed Reflexive Riesz Smith Stereotype Strictly convex Webbed Topological tensor product (of Hilbert spaces)
Mapping topologies	Dual Dual space Operator Ultraweak Weak (operator) Mackey Strong (polar operator) Ultrastrong Uniform convergence
Linear operators	Adjoint Bilinear (form) Bounded / Unbounded Closed Continuous / Discontinuous Compact 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 Open mapping Parseval's identity Schauder fixed-point
Analysis	Bochner space Differentiation in Fréchet spaces Derivatives (Fréchet Gâteaux functional holomorphic) Integrals (Bochner Dunford Gelfand–Pettis regulated weak) Functional calculus (Borel continuous holomorphic) Inverse function theorem (Nash–Moser theorem) Vector measure 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

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

Banach lattice

Examples and constructions

See also