Lp sum

In mathematics, and specifically in functional analysis, the L^p sum of a family of Banach spaces is a way of turning a subset of the product set of the members of the family into a Banach space in its own right. The construction is motivated by the classical L^p spaces.

Definition

Let $(X_{i})_{i \in I}$ be a family of Banach spaces, where $I$ may have arbitrarily large cardinality. Set $P := \prod_{i \in I} X_{i},$ the product vector space.

The index set $I$ becomes a measure space when endowed with its counting measure (which we shall denote by $μ$ ), and each element $(x_{i})_{i \in I} \in P$ induces a function $I \to ℝ, i \mapsto ‖ x_{i} ‖ .$

Thus, we may define a function $Φ : P \to ℝ \cup {\infty}, (x_{i})_{i \in I} \mapsto \int_{I} ‖ x_{i} ‖^{p} d μ (i)$ and we then set $⨁^{p} \underset{i \in I}{} X_{i} := {(x_{i})_{i \in I} \in P ∣ Φ ((x_{i})_{i \in I}) < \infty}$ together with the norm $‖ (x_{i})_{i \in I} ‖ := {(\int_{i \in I} ‖ x_{i} ‖^{p} d μ (i))}^{1 / p} .$

The result is a normed Banach space, and this is precisely the L^p sum of $(X_{i})_{i \in I} .$

Properties

Whenever infinitely many of the $X_{i}$ contain a nonzero element, the topology induced by the above norm is strictly in between product and box topology.
Whenever infinitely many of the $X_{i}$ contain a nonzero element, the L^p sum is neither a product nor a coproduct.

Banach space topics
Types of Banach spaces	Asplund Banach list Banach lattice Grothendieck Hilbert Inner product space Polarization identity (Polynomially) Reflexive Riesz L-semi-inner product (B Strictly Uniformly) convex Uniformly smooth (Injective Projective) Tensor product (of Hilbert spaces)
Banach spaces are:	Barrelled Complete F-space Fréchet tame Locally convex Seminorms/Minkowski functionals Mackey Metrizable Normed norm Quasinormed Stereotype
Function space Topologies	Banach–Mazur compactum Dual Dual space Dual norm Operator Ultraweak Weak polar operator Strong polar operator Ultrastrong Uniform convergence
Linear operators	Adjoint Bilinear form operator sesquilinear (Un)Bounded Closed Compact on Hilbert spaces (Dis)Continuous Densely defined Fredholm kernel operator Hilbert–Schmidt Functionals positive Pseudo-monotone Normal Nuclear Self-adjoint Strictly singular Trace class Transpose Unitary
Operator theory	Banach algebras C-algebras Operator space Spectrum C-algebra radius Spectral theory of ODEs Spectral theorem Polar decomposition Singular value decomposition
Theorems	Anderson–Kadec Banach–Alaoglu Banach–Mazur Banach–Saks Banach–Schauder (open mapping) Banach–Steinhaus (Uniform boundedness) 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 Parseval's identity Riesz's lemma Riesz representation Robinson-Ursescu Schauder fixed-point Sobczyk's theorem
Analysis	Abstract Wiener space Banach manifold bundle Bochner space Convex series Differentiation in Fréchet spaces Derivatives Fréchet Gateaux functional holomorphic quasi Integrals Bochner Dunford Gelfand–Pettis regulated Paley–Wiener weak Functional calculus Borel continuous holomorphic Measures Lebesgue Projection-valued Vector Weakly / Strongly measurable function
Types of sets	Absolutely convex Absorbing Affine Balanced/Circled Bounded Convex Convex cone (subset) Convex series related ((cs, lcs)-closed, (cs, bcs)-complete, (lower) ideally convex, (Hx), and (Hwx)) Linear cone (subset) Radial Radially convex/Star-shaped Symmetric Zonotope
Subsets / set operations	Affine hull (Relative) Algebraic interior (core) Bounding points Convex hull Extreme point Interior Linear span Minkowski addition Polar (Quasi) Relative interior
Examples	Absolute continuity AC $b a (Σ)$ c space Banach coordinate BK Besov $B_{p, q}^{s} (ℝ)$ Birnbaum–Orlicz Bounded variation BV Bs space Continuous C(K) with K compact Hausdorff Hardy H^p Hilbert H Morrey–Campanato $L^{λ, p} (Ω)$ ℓ^p $ℓ^{\infty}$ L^p $L^{\infty}$ weighted Schwartz $S (ℝ^{n})$ Segal–Bargmann F Sequence space Sobolev W^k,p Sobolev inequality Triebel–Lizorkin Wiener amalgam $W (X, L^{p})$
Applications	Differential operator Finite element method Mathematical formulation of quantum mechanics Ordinary Differential Equations (ODEs) Validated numerics

Functional analysis (topics – glossary)

Spaces

Banach Besov Fréchet Hilbert Inner product Normed Nuclear Schwartz Sobolev Topological vector
Properties	Barrelled Quasi-barrelled Complete Dual (Algebraic – Topological) Locally convex Reflexive polynomially Separable

Theorems

Operators

Algebras

Open problems

Applications

Advanced topics

Category