Real Analysis |
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Page 177
... convex if we can find a base for the topology consisting of convex sets . The following lemma is useful : 20. Lemma : Let O be an open set in a topological vector space . Then each point of O is internal . ф Proof : Let xo ε O. The ...
... convex if we can find a base for the topology consisting of convex sets . The following lemma is useful : 20. Lemma : Let O be an open set in a topological vector space . Then each point of O is internal . ф Proof : Let xo ε O. The ...
Page 178
... convex topological vector space , provided there are enough functionals in the family to separate the points of X ... set N which contains but does not meet F. Let 0 = N ( −N ) . Then O is an open convex set containing 0 and disjoint ...
... convex topological vector space , provided there are enough functionals in the family to separate the points of X ... set N which contains but does not meet F. Let 0 = N ( −N ) . Then O is an open convex set containing 0 and disjoint ...
Page 283
... set , 3 Equicontinuous family , 153 Equivalence , 17 Equivalent metrics ... convex , 177 LP spaces , 93 , 201 isometries of , 273 linear functionals on ... set , 47 , 217 , 245 Measurable space , 191 Metric space , 109 Minkowski ...
... set , 3 Equicontinuous family , 153 Equivalence , 17 Equivalent metrics ... convex , 177 LP spaces , 93 , 201 isometries of , 273 linear functionals on ... set , 47 , 217 , 245 Measurable space , 191 Metric space , 109 Minkowski ...
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A₁ absolutely continuous axiom B₁ Baire set Banach space Borel equivalent Borel sets Borel subset bounded linear functional called Cauchy sequence closed sets cluster point Co(X compact Hausdorff space compact space continuous function continuous real-valued functions convex set Corollary countable collection Daniell integral definition denote E₁ E₂ elements finite measure finite number following proposition function defined function f ƒ and g given Hausdorff space Hence homeomorphism infinite L₁ Lebesgue measure Lemma Let f Let ƒ linear manifold measurable function measurable sets measure algebra measure space measure zero metric space monotone natural numbers nonempty nonnegative measurable function o-algebra o-finite one-to-one open intervals open set outer measure point of closure Problem Proof Prove rational numbers semicontinuous set function set of finite set of measure Show simple function topological space unique vector lattice x₁