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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 1993 • Question 2

Question

Math 6210 • Real Analysis Fall 1993 • Question 2

Let \(X=[a,b]\subset \mathbb R\) be a closed interval. Assume \(a<b\). Regard \(X\) as a measure space with respect to the usual Lebesgue measure.

(a) Show directly that the space \(C(X)\) of continuous functions on \(X\) is not dense in \(L^\infty (X)\).

(b) Using (a), if necessary, show that there exists a bounded linear functional on \(L^\infty (X)\) which vanishes on \(C(X)\).

Raw LaTeX

Let \(X=[a,b]\subset\mathbb R\) be a closed interval. Assume \(a<b\).
Regard \(X\) as a measure space with respect to the usual Lebesgue measure.

(a) Show directly that the space \(C(X)\) of continuous functions on \(X\) is not
dense in \(L^\infty(X)\).

(b) Using (a), if necessary, show that there exists a bounded linear functional
on \(L^\infty(X)\) which vanishes on \(C(X)\).

Topic Tags

Bounded Linear Operators Dual Spaces and Hahn-Banach Theorem Lebesgue Measure and Translation Invariance Lp Spaces and Sequence Spaces