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    "title": "Kakutani's direct common-fixed-point proof and congruent-group generalization",
    "summary": "Kakutani gave a direct averaging proof of Markov's common-fixed-point theorem for a commuting family of continuous affine self-maps of a nonempty compact convex subset of a locally convex topological vector space. He also formulated a common-fixed-point theorem for an affine congruent group preserving such a set, with applications to almost-periodic means and an alternative route to Banach's extension theorem.",
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        "description": "Compactness, convexity, local convexity and affine action are essential stated hypotheses. The commuting-family theorem does not assume surjectivity; the solvable-group corollary and congruent-group theorem require bijective preservation of the compact set. Congruence means preservation of a common zero-neighborhood system, not arbitrary affine groups. The note contains statements and brief proof summaries, explicitly deferring details. It does not establish uniqueness of the mean under its general hypotheses. The note explicitly supplies statements and brief proof summaries, leaving details elsewhere. Compact convex nonempty set in a locally convex space; commuting self-maps need not be surjective, but the group variants preserve the set bijectively. Congruence is not an arbitrary group property and uniqueness of a mean does not follow. Not the separate1941 set-valued fixed-point theorem.",
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      "statement": "# Kakutani's direct common-fixed-point proof and congruent-group generalization\n\nKakutani gave a direct averaging proof of Markov's common-fixed-point theorem for a commuting family of continuous affine self-maps of a nonempty compact convex subset of a locally convex topological vector space. He also formulated a common-fixed-point theorem for an affine congruent group preserving such a set, with applications to almost-periodic means and an alternative route to Banach's extension theorem.\n\n## Scientific scope\n\nCompactness, convexity, local convexity and affine action are essential stated hypotheses. The commuting-family theorem does not assume surjectivity; the solvable-group corollary and congruent-group theorem require bijective preservation of the compact set. Congruence means preservation of a common zero-neighborhood system, not arbitrary affine groups. The note contains statements and brief proof summaries, explicitly deferring details. It does not establish uniqueness of the mean under its general hypotheses. The note explicitly supplies statements and brief proof summaries, leaving details elsewhere. Compact convex nonempty set in a locally convex space; commuting self-maps need not be surjective, but the group variants preserve the set bijectively. Congruence is not an arbitrary group property and uniqueness of a mean does not follow. Not the separate1941 set-valued fixed-point theorem.\n\n## Reading status\n\nThis is a source-linked WRS catalogue note. A full explanatory guide and independent scientific reconstruction have not been completed for this entry. Read the identified original work for its complete narrative and arguments.\n\n## People and systems\n\nShizuo Kakutani\n\nSource-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred.\n\n\n\n## Original sources\n\n- [Two Fixed-point Theorems Concerning Bicompact Convex Sets](https://www.jstage.jst.go.jp/article/pjab1912/14/7/14_7_242/_pdf/-char/en)\n",
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      "title": "Kakutani's direct common-fixed-point proof and congruent-group generalization"
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    "scope": "Kakutani gave a direct averaging proof of Markov's common-fixed-point theorem for a commuting family of continuous affine self-maps of a nonempty compact convex subset of a locally convex topological vector space. He also formulated a common-fixed-point theorem for an affine congruent group preserving such a set, with applications to almost-periodic means and an alternative route to Banach's extension theorem. Compactness, convexity, local convexity and affine action are essential stated hypotheses. The commuting-family theorem does not assume surjectivity; the solvable-group corollary and congruent-group theorem require bijective preservation of the compact set. Congruence means preservation of a common zero-neighborhood system, not arbitrary affine groups. The note contains statements and brief proof summaries, explicitly deferring details. It does not establish uniqueness of the mean under its general hypotheses. The note explicitly supplies statements and brief proof summaries, leaving details elsewhere. Compact convex nonempty set in a locally convex space; commuting self-maps need not be surjective, but the group variants preserve the set bijectively. Congruence is not an arbitrary group property and uniqueness of a mean does not follow. Not the separate1941 set-valued fixed-point theorem.",
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