### Multi-valued Namioka theorems.

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The theorem in the title is proven. Applications to product theorems are given.

Let ${x}_{0}$ be a q-point of a regular space $X,Y$ a Hausdorff space whose relatively countably compact subsets are relatively compact and let $F:X\to Y$ be an upper semicontinuous set valued map. Then the active boundary $FracF\left({x}_{0}\right)$ is the smallest compact kernel of $F$ at ${x}_{0}$.

It is shown that the proper domains of integral operators have separating duals but in general they are not locally convex. Banach function spaces which can occur as proper domains are characterized. Some known and some new results are given, illustrating the usefulness of the notion of proper domain.

The area of research of this paper goes back to a 1930 result of H. Auerbach showing that a scalar series is (absolutely) convergent if all its zero-density subseries converge. A series ${\sum}_{n}{x}_{n}$ in a topological vector space X is called ℒ-convergent if each of its lacunary subseries ${\sum}_{k}{x}_{{n}_{k}}$ (i.e. those with ${n}_{k+1}-{n}_{k}\to \infty $) converges. The space X is said to have the Lacunary Convergence Property, or LCP, if every ℒ-convergent series in X is convergent; in fact, it is then subseries convergent. The Zero-Density Convergence...

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