2. In real analysis, or functional analysis, please briefly show your work explicitly and clearly. 2. Which of the following sequences (if any) are uniformly convergent on R? equicontinuous? a) xn= exp(-nt2) b) yn = exp(-(t - n)2) c) zn = 1 cos n2t.
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Step 1: We will first check for uniform convergence of the sequences. Show more…
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Consider the following sequence of functions: fn(x) = nx^2 / (n + x) (a) Find the pointwise limit of the sequence of functions (fn) on [0, ∑). Briefly explain your reasoning. (b) Show that the sequence of functions (fn) is not convergent uniformly on [0, ∑). (c) Let a < b, a, b ∈ [0, ∑). Prove that the sequence (fn) is convergent uniformly on [a, b].
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(c) [i] Suppose (fn) and (gn) are two sequences of functions that converge uniformly on a subset E ⊂ ℑ. Is it true that the sequence (fngn) converges uniformly on E? Prove or give a counter example. [ii] If your answer to [i] was “YES”, does the result still hold if one of the sequences does not converge uniformly? If your answer to [i] was “NO”, what additional assumption on (fn) and (gn) will yield uniform convergence of (fngn) ?
(a) If {fn} and {gn} converge uniformly on a set E, prove that {fn + gn} converges uniformly on E. (b) If in addition {fn} and {gn} are sequences of bounded functions, prove that {fn gn} converges uniformly on E. (c) Construct sequences {fn}, {gn} which converge uniformly on some set E, but such that {fngn} does not converge uniformly on E.
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