Gelfand-Pettis integrals and weak holomorphy
نویسنده
چکیده
If X is a vector space and E is a subset of X, the convex hull of E is defined to be the intersection of all convex sets containing E, and is denoted by co(E). One checks that the convex hull of E is equal to the set of all finite convex combinations of elements of E. If X is a topological vector space, the closed convex hull of E is the intersection of all closed convex sets containing E, and is denoted by co(E). The closed convex hull of E is equal to the closure of the convex hull of E.
منابع مشابه
Vector-valued integrals
Quasi-complete, locally convex topological vector spaces V have the useful property that continuous compactly-supported V -valued functions have integrals with respect to finite Borel measures. Rather than constructing integrals as limits following [Bochner 1935], [Birkhoff 1935], et alia, we use the [Gelfand 1936][Pettis 1938] characterization of integrals, which has good functorial properties...
متن کاملPreview of vector-valued integrals
In contrast to construction of integrals as limits of Riemann sums, the Gelfand-Pettis characterization is a property no reasonable notion of integral would lack. Since this property is an irreducible minimum, this definition of integral is called a weak integral. Uniqueness of the integral is immediate when the dual V ∗ separates points, meaning that for v 6 v′ in V there is λ ∈ V ∗ with λv 6=...
متن کاملThe incompleteness of weak duals
1. Incompleteness of weak duals of reasonable spaces 2. Appendix: locally-convex limits and colimits 3. Appendix: ubiquity of quasi-completeness The point here is to prove that the weak duals of reasonable topological vector spaces, such as infinite-dimensional Hilbert, Banach, or Fréchet spaces, are not complete. That is, in these weak duals there are Cauchy nets which do not converge. Happily...
متن کاملA Note on Weak Differentiability of Pettis Integrals
Pettis raised the question whether or not separability of the range space implies almost everywhere weak differentiability of Pettis integrals. Phillips has given an example which answers this question in the negative. His construction is based on a sequence of orthogonal vectors in Hilbert space. We present here a different example of the same type of function. Our basic construction is that o...
متن کاملHarmonic analysis on compact abelian groups
Paul Garrett [email protected] http://www.math.umn.edu/ g̃arrett/ [This document is http://www.math.umn.edu/ ̃garrett/m/fun/notes 2012-13/06c cpt ab gps.pdf] 1. Approximate identities on topological groups 2. Uniqueness of invariant measure 3. Simultaneous eigenfunctions for integral operators 4. Simultaneous eigenfunctions are characters The spectral theory for normal compact operators on Hil...
متن کامل