Dominated Convergence and Stone-Weierstrass Theorem
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J.-P. Jurzak
Abstract
Let C(X; ℝ) the algebra of continuous real valued functions defined on a locally compact space X. We consider linear subspaces 𝔸 ⊂ C(X; ℝ) having the following property: there is a sequence (Φj)j∈ℕ of positive functions in 𝔸 with limx→∞ Φj(x) = +∞ for every j ∈ ℕ, such that 𝔸 consists of functions ƒ ∈ C(X; ℝ) bounded above for the absolute value by an homothetic of some Φn (n depends on each ƒ). Dominated convergence of a sequence (gn)n≥1 in 𝔸 is an estimation of the form |gn(x) – g(x)| ≤ εn|h(x)| for all x ∈ X and all n ∈ ℕ where gn, g, h ∈ 𝔸 and εn → 0 as n → ∞. We extend the Stone-Weierstrass theorem to subalgebras or lattices 𝔹 ⊂ 𝔸 and we show that the dominated convergence for sequences is exactly the convergence of sequences when 𝔸 is endowed with a locally convex (DF)-space topology.
© Heldermann Verlag
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- Projections in Weakly Compactly Generated Banach Spaces and Chang's Conjecture
- Dominated Convergence and Stone-Weierstrass Theorem
- Optimality Conditions and Duality for Multiobjective Control Problems
- On the Convergence of Sequences of Functions which are Discontinuous on Countable Sets
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Articles in the same Issue
- On Additive Almost Continuous Functions Under
- Maximal Solutions and Existence Theory for Fuzzy Differential and Integral Equations
- Projections in Weakly Compactly Generated Banach Spaces and Chang's Conjecture
- Dominated Convergence and Stone-Weierstrass Theorem
- Optimality Conditions and Duality for Multiobjective Control Problems
- On the Convergence of Sequences of Functions which are Discontinuous on Countable Sets
- Euler-Poincaré Formalism of Coupled KdV Type Systems and Diffeomorphism Group on S1
- Stability Analysis and Comparison of the Models for Carcinogenesis Mutations in the Case of Two Stages of Mutations
