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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2008 • Question 5

Question

Math 6210 • Real Analysis Spring 2008 • Question 5

Let \(\ell ^\infty \) denote the Banach space of bounded real sequences with norm \(\lVert x\rVert _\infty =\sup _i |x_i|\), and \(\ell ^1\) the Banach space of real sequences with norm \(\lVert x\rVert _1=\sum _i |x_i|<\infty \). For \(x\in \ell ^\infty \) and \(y\in \ell ^1\) define \(Tx(y)=\sum _i x_i y_i\). Prove that \(T:\ell ^\infty \to (\ell ^1)^*\) is a bounded operator that is one to one and onto. (Here \((\ell ^1)^*\) is the dual space of \(\ell ^1\).)

Raw LaTeX

Let \(\ell^\infty\) denote the Banach space of bounded real
sequences with norm \(\lVert x\rVert_\infty=\sup_i |x_i|\), and \(\ell^1\) the
Banach space of real sequences with norm
\(\lVert x\rVert_1=\sum_i |x_i|<\infty\). For \(x\in\ell^\infty\) and
\(y\in\ell^1\) define \(Tx(y)=\sum_i x_i y_i\). Prove that
\(T:\ell^\infty\to(\ell^1)^*\) is a bounded operator that is one to one and
onto. (Here \((\ell^1)^*\) is the dual space of \(\ell^1\).)

Topic Tags

Banach Spaces Dual Spaces and Hahn-Banach Theorem Lp Spaces and Sequence Spaces