Is $T(X)$ a Banach space?

Question

Let $X$ be a Banach space and $Y$ be a normed space and $T \colon X\to Y$ be a linear transformation. Then is $T(X)$ a Banach space? How do I prove this? I'm asking this since I saw a post saying this is true some hours ago (I don't remember the reference though, maybe it was on some forum and I believe the statement is false).

EDIT:

Let $T:V\to W$ is a bounded bijective operator where $V$ is a Banach space and $W$ is a normed space. Then, is $T(V)$ a Banach space?

The range of a bounded linear transformation need not be closed (see this, e.g.). It's false. — David Mitra, Jan 01 '15 at 17:35
@DavidMitra Hmm.. i'm curious what condition should $T$ have to make $T(X)$ complete. What if $T$ is a bounded bijection? I'm really curious about it so I just edited my post. — Rubertos, Jan 01 '15 at 17:59

score 5 · Accepted Answer · edited Apr 13 '17 at 12:21

5

No. Let $T:\ell^2\to \ell^2$ be defined by $T(x_1,x_2,x_3,\ldots)=\left(x_1,\frac12x_2,\frac13x_3,\ldots\right)$. Then $T$ is injective and bounded, but its range is not closed. Let $X=\ell^2$ and $Y=T(\ell^2)$ to get a counterexample for your question with $T$ being a bounded bijection from a Banach space to a normed space.

For your more general question in a comment, "I'm curious what condition should $T$ have to make $T(X)$ complete," in the context of $Y$ being a Banach space, see When is the image of a linear operator closed?

edited Apr 13 '17 at 12:21

Community

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answered Jan 01 '15 at 18:12

Jonas Meyer

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Thank you so much :) especially for the link – Rubertos Jan 01 '15 at 18:24

Is $T(X)$ a Banach space?

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