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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

2000 • Question 3

Question

Math 6210 • Real Analysis 2000 • Question 3

Let \(X=[0,1]\), and \(C(X)\) the space of complex valued continuous functions on \(X\), equipped with the sup norm: \(\|f\|_\infty =\sup _{x\in X}|f(x)|\). Let \(f_n\) be a sequence in \(C(X)\). Prove that the following statements are equivalent:

(a) For each bounded linear functional \(\lambda \) on \(C(X)\), \(\lambda (f_n)\to 0\);

(b) \(f_n(x)\to 0\) for each \(x\in X\), and \(\sup _n\|f_n\|_\infty \) is finite.

Raw LaTeX

Let \(X=[0,1]\), and \(C(X)\) the space of complex valued continuous
functions on \(X\), equipped with the sup norm:
\(\|f\|_\infty=\sup_{x\in X}|f(x)|\). Let \(f_n\) be a sequence in \(C(X)\). Prove
that the following statements are equivalent:

(a) For each bounded linear functional \(\lambda\) on \(C(X)\), \(\lambda(f_n)\to 0\);

(b) \(f_n(x)\to 0\) for each \(x\in X\), and \(\sup_n\|f_n\|_\infty\) is finite.

Topic Tags

Almost Everywhere and Convergence in Measure Bounded Linear Operators Dual Spaces and Hahn-Banach Theorem Weak Topologies and Weak Convergence