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Suppose $T:X\rightarrow Y$ is a continuous linear map of Banach spaces, say. Let $D$ be a dense subspace of $X$ and assume $T$ is injective on $D$. Does it follow that $T$ is injective? I would guess not, but what is a counterexample?

Does this work if instead $X$, $Y$ are Hilbert Spaces? (I've tried some maps on $l^2$)

Jeff
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  • I'm sorry. I deleted my answer because I realized I massively misread your question. I thought you were interested in continuously extending $T$ to $X$ for some reason. – Stella Biderman Mar 19 '14 at 06:27

1 Answers1

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Let $H=\ell^2(\mathbb{N}),\ \{e_k,\ k\in\mathbb{N}\}$ be the standard base. Define the projection operator $$A\big((x_1,x_2,x_3,\dots)\big) = (0,x_2,x_3,\dots),$$ and $$D = \operatorname{lin}\{f_1,e_2,e_3,\dots\},$$ where $f_1=(1,\frac 1 2,\frac 1 3,\dots)$. Then $D$ is dense, $e_1\notin D$, so $A$ is injective on $D$ but not on $H$.

Nathanael Skrepek
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Yurii Savchuk
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