By Jean Francois Colombeau
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Additional info for Differential Calculus and Holomorphy: Real and complex analysis in locally convex spaces
Example text
A μ-measurable mapping F : X → E is stochastically Gˆateaux differentiable if there exists a measurable mapping D H F : X → H(H, E) such that for every h ∈ H we have F(x + th) − F(x) − DH F(x)(h) −−→ 0 in measure μ. t→0 t The derivative of the n th order D nH F is deVned inductively as D H (D n−1 H F). An alternative notation is ∇ Hn F. 3. Let 1 ≤ p < ∞. The space D p,1 (μ, E) is deVned as the class of all mappings f ∈ L p (μ, E) such that f is ray absolutely continuous, stochastically Gˆateaux differentiable and D H f ∈ L p μ, H(H, E) .
Var. Partial Diff. Equ. 45 (2012), n. 1-2, 101–124. [29] P. C ELADA and A.
The corresponding space of equivalence classes will be denoted by the same symbol. This is the exact analog of the class M H (μ) in Section 8. 1. The set M H (U, μ) is a Banach space with the norm f M := f L 1 (U,μ) + sup |h| H ≤1 fβhμ L 1 (U,μ) . Proof. Let us observe that the operator h → fβh from H to L 1 (U, μ) is linear and has a closed graph. Indeed, suppose that h n → h in H and fβh n → g in L 1 (U, μ). By the continuity of the embedding H → D(μ) we have βh n → βh in L 1 (μ), whence it follows that fβh n → fβh in measure on U , hence g = fβh .