Prove C(x) with the supremum norm is a metric space by showing the supremum norm is a metric. Suppose {fn}n=1 converges uniformly on E. Show {fn}n=1 converges pointwise on E. Let {fn}n=1 and {gn}n=1 converges uniformly on E. Prove that {fn - gn}n=1 converge uniformly on F
Prove C(x) with the supremum norm is a metric space by showing the supremum norm is a metric. Suppose {fn}n=1 converges uniformly on E. Show {fn}n=1 converges pointwise on E. Let {fn}n=1 and {gn}n=1 converges uniformly on E. Prove that {fn - gn}n=1 converge uniformly on F
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:1. Prove C(x) with the supremum norm is a metric space by showing the supremum norm
is a metric.
2. Suppose {fn}n=1 converges uniformly on E. Show {fnn-1 converges pointwise on E.
3. Let {fn}n=1 and {gn}-1 converges uniformly on E. Prove that {fn - gn}n=1 Converge
uniformly on E.
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Step 1
Uniform convergence implies the pointwise convergence of the sequence.
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