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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2011 • Question 1

Question

Math 6210 • Real Analysis Spring 2011 • Question 1

Recall that a topological space is said to be separable if it contains a countable dense set. Also recall that \(\ell ^p(\mathbb Z)\) is the \(L^p\)-space for counting measure on the integers. Prove or disprove the following statements:

(a) \(\ell ^1(\mathbb Z)\) is separable.

(b) \(\ell ^\infty (\mathbb Z)\) is separable.

Raw LaTeX

Recall that a topological space is said to be separable if it
contains a countable dense set. Also recall that \(\ell^p(\mathbb Z)\) is the
\(L^p\)-space for counting measure on the integers. Prove or disprove the
following statements:

(a) \(\ell^1(\mathbb Z)\) is separable.

(b) \(\ell^\infty(\mathbb Z)\) is separable.

Topic Tags

Absolute Continuity and Singular Measures Compactness in Metric and Normed Spaces Lp Spaces and Sequence Spaces Measure Theory