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algebraically closed Archimedean assume Banach space BC(X BUC(X C-algebra choose closed linear hull closed linear subspace compact open subgroup compact semigroup compact subgroup contains COROLLARY cosets countable type covering ring define denote dense dimensional discrete equivalent EXERCISE finite formula group G Haar measure Hausdorff space Hence idempotent IIall IITII induces isometry isomorphism KIIXII left Haar measure left invariant measure LEMMA Let G linear hull linear isometry linear subspace llxll locally compact group locally compact zerodimensional maximal ideal natural map non-Archimedean non-Archimedean uniformity non-measurable normed vector space orthogonal base orthogonal system orthonormal polynomial PROOF prove q-free q-free compact quotient reflexive residue class field semigroup sequence spherically complete subgroup H subgroup of G subgroup topology subset surjective THEOREM theory tight measures topological group topological space torsional group trivial ultrametric ultrametrizable unit element valuation value group zerodimensional Hausdorff space