Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Extension Of The Bauer's Maximum Principle For Compact Metrizable Sets   Let X be a nonempty convex compact subset of some Haus-dorff locally convex
topological vector space S. The well know Bauer's maximum principle stats that
every convex upper semi-continuous function from X into R attains its maximum
at some extremal point of X. We give some extensions of this result when X is
assumed to be compact metrizable. We prove that the set of all convex upper
semi-continuous functions attaining there maximum at exactly one extremal point
of X is a G $\delta$ dense subset of the space of all convex upper
semi-continuous functions equipped with a metric compatible with the uniform
convergence .

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The categories of this paper are: comp-gas
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The categories of this paper are: chao-dyn
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The categories of this paper are: hep-lat
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The categories of this paper are: q-alg
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The categories of this paper are: gr-qc
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The categories of this paper are: econ
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The categories of this paper are: nucl-th
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The categories of this paper are: quant-ph
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The categories of this paper are: supr-con
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The categories of this paper are: nlin
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The categories of this paper are: stat
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The categories of this paper are: math-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: q-fin