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Strongly exposed points in Orlicz spaces of vector-valued functions. I. (English) Zbl 0639.46029
If Y is a real Banach space with dual $$Y^*$$, and K is a nonempty bounded subset of Y, a point $$y_ 0$$ in K is called strongly exposed if there is an element $$y^*$$ in $$Y^*$$ such that (i) $$y^*(y_ 0)=\sup y^*(K)$$, and (ii) for each sequence $$(y_ n)$$ in K with $$\lim_{n\to \infty}y^*(y_ 0-y_ n)=0$$ we have $$\lim_{n\to \infty}\| y_ 0- y_ n\| =0$$. It is shown that strongly exposed functions in the Orlicz spaces $$L_{\Phi}(\mu,X)$$ with $$\Phi =\phi \circ | \cdot |_ X$$ can be characterized in terms of the same property for their values in X. In the case of Bochner $$L_ p$$ spaces, $$1<p<\infty$$, the smoothness condition assumed in P. Greim, Proc. Am. Math. Soc. 88, 81-84 (1983; Zbl 0524.46021), is removed. A space is said to have the strongly exposing property if every point of the unit sphere is strongly exposed. The latter part of the paper studies this property for the space $$L_{\phi}(\mu,{\mathbb{R}})$$.
Reviewer: G.A.Heuer

##### MSC:
 4.6e+31 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 4.6e+41 Spaces of vector- and operator-valued functions
Zbl 0524.46021