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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2024 • Question 6

Question

Math 6210 • Real Analysis Fall 2024 • Question 6

Prove that the dual space of \((c_0,\|\cdot \|_{\sup })\) is \(\ell ^1\). Recall \[ c_0=\{v\in \mathbb {R}^{\mathbb {N}}:\lim _{j\to \infty }v_j=0\} \]

and \(\ell ^1=\{w\in \mathbb {R}^{\mathbb {N}}:\sum _j |w_j|<\infty \}\) with \(\|w\|=\sum _j |w_j|\).

Raw LaTeX

Prove that the dual space of \((c_0,\|\cdot\|_{\sup})\) is \(\ell^1\). Recall
\[
c_0=\{v\in\mathbb{R}^{\mathbb{N}}:\lim_{j\to\infty}v_j=0\}
\]

and \(\ell^1=\{w\in\mathbb{R}^{\mathbb{N}}:\sum_j |w_j|<\infty\}\) with \(\|w\|=\sum_j |w_j|\).

Topic Tags

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