What is the dual space of $c_{00}$?

Asked Dec 19 '24 at 08:54

Active Dec 19 '24 at 08:54

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Consider $c_{00}$ (Space of sequences that has finite nonzero elements) equipped with norm $\|\cdot\|_{\infty}$. Its algebraic dual space $\mathbb{C}^\infty$ is mentioned in this answer. However it can't be dual space as functional $f\in \mathbb{C}^\infty$ can be unbounded.

I only know $(c_{00})^* \supset \ell^1$ because $(c_0)^* = \ell^1$. I don't know $(c_{00})^* = \ell^1$ or the dual space would be larger.

asked Dec 19 '24 at 08:54

Zequan Bear

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Its closure is $c_0$ so its dual is same as that of $c_0$, namely $\ell^{1}$. – Kavi Rama Murthy Dec 19 '24 at 08:56
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Does it mean the space and the closure of this space must have the same dual space? Looks like it comes from the continuity of functional. – Zequan Bear Dec 19 '24 at 09:24
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Any continuous linear functional on a subspace has a unique norm preserving extension to the closure of the subspace, so the duals are the same. – Kavi Rama Murthy Dec 19 '24 at 09:26
Where does the "unique" come from? By Hahn-Banach theorem we can only get the existence. Thank you for your help. – Zequan Bear Dec 19 '24 at 14:57
Hahn-Banach Theorem is not required. Uniqueness is just a basic Topology argument. – Kavi Rama Murthy Dec 19 '24 at 22:50

What is the dual space of $c_{00}$?

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