By S. G. Vladut
During the second one half the nineteenth century Leopold Kronecker loved a dream, his Jugendtraum, that he may still see the formula of a whole concept of advanced multiplication. during this vital ebook Serge Vladut has studied this dream, tracing the improvement of elliptic functionality concept from its genesis to themost fresh achievements.
Read or Download Kronecker's Jugendtraum and Modular Functions (Studies in the Development of Modern Mathematics) PDF
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Additional info for Kronecker's Jugendtraum and Modular Functions (Studies in the Development of Modern Mathematics)
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 .