Let $X,Y$ be normed spaces over $\mathbb C$. A linear map $T\colon X\to Y$ is compact if $T$ carries bounded sets into relatively compact sets (i.e sets with compact closure). Equivalently if $x_n\in X$ is a bounded sequence, there exist a subsequence $x_{n_k}$ such that $Tx_{n_k}$ is convergent. I want to prove that if $T\colon X\to Y$ has finite dimensional range, then it's compact.
$\begingroup$
$\endgroup$
1
-
11$\begingroup$ That's only true if $T$ is continuous. $\endgroup$– Daniel FischerMar 11, 2016 at 20:32
Add a comment
|
5 Answers
$\begingroup$
$\endgroup$
3
Here is a direct proof.
Proof. Since $T$ has finite rank, Im$T$ is a finite-dimensional normed space. Furthermore, for any bounded sequence $\{ x_n \}$ in $X$, the sequence $\{ T x_n \}$ is bounded in Im$T$, so by Bolzano-Weierstrass theorem this sequence must contain a convergent subsequence. Hence $T$ is compact. $\square$
-
$\begingroup$ for any bounded sequence ${x_n}$ in$ X$, the sequence ${Tx_n}$ is bounded in $ImT$. Why can we say that? $\endgroup$– LucasJun 15, 2019 at 20:15
-
1$\begingroup$ The usual definition of finite rank operator is the following: a linear map $T: X \to Y$ ($X$, $Y$ normed linear spaces) is a finite rank operator if and only if $T$ is bounded and $\mbox{dim} \mbox{Im} T < \infty$. Thus, if $T$ is finite rank, $T$ is bounded. But it is also linear, therefore $T(A)$ is bounded for any bounded $A \subset X$. (Note that this property, as well as boundedness, is equivalent to continuity for linear maps, and continuity is essential, as pointed out by Daniel Fischer just below the OP.) $\endgroup$ Jun 15, 2019 at 20:44
-
$\begingroup$ T has to be a bounded operator as nothing guarantees that $Tx_n$ is bounded. $\endgroup$ Mar 8, 2021 at 17:38
$\begingroup$
$\endgroup$
Hint: Every finite dimensional normed space is isomorphic to $\mathbb R^n$, so any bounded subset is pre-compact (sets with compact closure).
$\begingroup$
$\endgroup$
if $T$ is bounded and dim rangeT is finite ,the operator $T$ is compact.(kreyszig p.407)
$\begingroup$
$\endgroup$
0
You require the boundedness of $T$. Fact is that boundedness + finite rank of $T$ gives compactness of $T$. This you can find in any good book in Functional Analysis. I would recommend J.B Conway
$\begingroup$
$\endgroup$
Assuming $T$ is continuous, you know that if $||x||=1$, then $T(x) \leq M$ for some $M \in \mathbb{R}$. So $T(B^{X}_{1}(0)$) (unit ball in $X$) is contained in $\overline{B^{\text{Im}(T)}_{M}(0)}$, which is compact since the image is finite dimensional and hence isomorphic to $\mathbb{R}^{n}$.
