Question: Let (ƒ) be a sequence of functions defined on a Banach space (X, || ⋅ ||), where each ƒ„ : X → R is a continuous linear functional. Suppose (fn) converges pointwise to a function ƒ : X → R, i.e., for every x Є X, limn→∞ ƒn(x) = f(x). Assume further that there exists a sequence of positive numbers (Mm) such that ||fn|| ≤ M₂ for all n, where M → M as n → ∞. 1. Prove that if the sequence (ƒ) is uniformly bounded, i.e., supm ||fn|| < ∞, then the pointwise limit f is a continuous linear functional on X. 2. Suppose instead that (ƒ) is a sequence of linear functionals converging to ƒ uniformly on every bounded subset of ✗. Prove that f is indeed a continuous linear functional on X, and show that lim→∞ || fn — f || = 0. 3. Given that (✗, || . ||) is a Hilbert space, explore the implications of the above results in the context of weak convergence. Specifically, if (f) converges weakly to f, does it follow that (fn) converges in norm to f? Provide a rigorous proof or counterexample.
Question: Let (ƒ) be a sequence of functions defined on a Banach space (X, || ⋅ ||), where each ƒ„ : X → R is a continuous linear functional. Suppose (fn) converges pointwise to a function ƒ : X → R, i.e., for every x Є X, limn→∞ ƒn(x) = f(x). Assume further that there exists a sequence of positive numbers (Mm) such that ||fn|| ≤ M₂ for all n, where M → M as n → ∞. 1. Prove that if the sequence (ƒ) is uniformly bounded, i.e., supm ||fn|| < ∞, then the pointwise limit f is a continuous linear functional on X. 2. Suppose instead that (ƒ) is a sequence of linear functionals converging to ƒ uniformly on every bounded subset of ✗. Prove that f is indeed a continuous linear functional on X, and show that lim→∞ || fn — f || = 0. 3. Given that (✗, || . ||) is a Hilbert space, explore the implications of the above results in the context of weak convergence. Specifically, if (f) converges weakly to f, does it follow that (fn) converges in norm to f? Provide a rigorous proof or counterexample.
Chapter3: Functions
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Transcribed Image Text:Question:
Let (ƒ) be a sequence of functions defined on a Banach space (X, || ⋅ ||), where each ƒ„ : X →
R is a continuous linear functional. Suppose (fn) converges pointwise to a function ƒ : X → R,
i.e., for every x Є X, limn→∞ ƒn(x) = f(x). Assume further that there exists a sequence of
positive numbers (Mm) such that ||fn|| ≤ M₂ for all n, where M → M as n → ∞.
1. Prove that if the sequence (ƒ) is uniformly bounded, i.e., supm ||fn|| < ∞, then the pointwise
limit f is a continuous linear functional on X.
2. Suppose instead that (ƒ) is a sequence of linear functionals converging to ƒ uniformly on
every bounded subset of ✗. Prove that f is indeed a continuous linear functional on X, and
show that lim→∞ || fn — f || = 0.
3. Given that (✗, || . ||) is a Hilbert space, explore the implications of the above results in the
context of weak convergence. Specifically, if (f) converges weakly to f, does it follow that
(fn) converges in norm to f? Provide a rigorous proof or counterexample.
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