L2valued

L2valued is a term used in functional analysis and probability to describe objects that take values in an L2 space. In practice, one often speaks of L2-valued random variables or L2-valued functions. The designation emphasizes that the values live in a space of square-integrable functions rather than in a finite-dimensional vector space.

In mathematical terms, let H be a separable Hilbert space and (Ω, F, P) a probability space. An H-valued random variable X: Ω → H is called L2-valued if E[||X||_H^2] < ∞. The Bochner space L2(Ω; H) consists of all such X, equipped with the norm (E[||X||^2])^{1/2}. There is a canonical isomorphism L2(Ω) ⊗ H ≅ L2(Ω; H), connecting scalar-valued and H-valued square-integrable objects. Measurability concepts for L2-valued objects include strong (Bochner) and weak measurability, which ensure that the Bochner integral is well defined.

Examples of L2-valued objects include random fields X: Ω → L2(D) with finite second moment, where for almost every ω, X(ω) is a square-integrable function on a spatial domain D. In functional data analysis and stochastic partial differential equations, solutions or data sets are often modeled as L2-valued random elements. L2-valued processes arise when a parameter (such as time) indexes a family of square-integrable functions.

Properties and operations for L2-valued objects mirror those for standard L2 spaces: linear combinations of L2-valued random variables remain L2-valued, and inner products are defined by E[⟨X(ω), Y(ω)⟩_H]. These objects support standard probabilistic and functional-analytic tools, including orthogonality, projections, and moment calculations, making them central to theories that handle function-valued data and random fields.