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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2020 • Question 5

Question

Math 6210 • Real Analysis Spring 2020 • Question 5

Let \(f_n\in C([0,1])\) for \(n=1,2,\ldots \). Show that the following statements are equivalent:

(a) For every \(\lambda \in C([0,1])^*\) we have \(\lambda (f_n)\to 0\) as \(n\to \infty \).

(b) \(f_n(x)\to 0\) for every \(x\in [0,1]\) and \(\sup \lVert f_n\rVert _\infty <\infty \).

Here \(\lVert f\rVert _\infty =\sup _{x\in [0,1]}|f(x)|\).

Raw LaTeX

Let \(f_n\in C([0,1])\) for \(n=1,2,\ldots\). Show that the following
statements are equivalent:

(a) For every \(\lambda\in C([0,1])^*\) we have \(\lambda(f_n)\to0\) as
\(n\to\infty\).

(b) \(f_n(x)\to0\) for every \(x\in[0,1]\) and
\(\sup\lVert f_n\rVert_\infty<\infty\).

Here \(\lVert f\rVert_\infty=\sup_{x\in[0,1]}|f(x)|\).

Topic Tags

Almost Everywhere and Convergence in Measure Weak Topologies and Weak Convergence