Elements of Real Analysis |
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algebra arbitrarily given assume Banach space bijective BM(E Boolean called Cartesian product Cauchy sequence closed sets closed subset completes the proof consider an arbitrary contains continuous function converges Daniell integral Define a function definition denote an arbitrary denote any point disjoint easily verify equivalent exists a natural exists a positive family F finite following corollary following proposition Fréchet space function f ƒ is continuous Hausdorff space Hence we obtain Hilbert space holds homeomorphism implies injective function inverse image isometric Lebesgue integrable LEMMA Let f Let X denote measurable function measurable subset measure space metric space natural number neighborhood nonempty nonnegative normed linear space open interval open set open subset orthonormal system positive real number pseudo-metric space pseudo-normed linear space real line remains to prove satisfying surjective function THEOREM topological group topological space topology vector