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$T$ is continuous if and only if $\ker T$ is closed

Let $T: X\to \mathbf{R}$ be linear. Suppose that $X$ is a Banach space. I want to show that $T$ is continuous if and only if $\ker T $ is closed.

My Attempt.

$(\Rightarrow)$ Suppose $T$ is continuous. Then if $x_n\to x$, then $T(x_n)\to T(x)$. Let $x_n \in \ker T$. Then $T(x_n) = 0$. Using continuity, $$ T(x) = \lim_{n\to \infty} T(x_n) = 0.$$ Hence $x\in \ker T$ and thus $\ker T$ is closed.

$(\Leftarrow)$ Suppose $T$ is not continuous. So $T$ is not bounded. i.e. $\exists$ a sequence $x_n$ such that $T(x_n) \to \infty$ as $n\to \infty$. Let $a\notin \ker T$. Then defining $$x_n' = a - \frac{T(a)}{T(x_n)}x_n ,$$ it is clear that $T(x_n') = 0$ and so $x_n'\in \ker T$. Also $x_n' \to a \notin \ker T.$ So $\ker T$ is not closed. Hence $\ker T$ closed implies that $T$ is continuous.

Have I approached this question correctly? Are there other ways of approaching it?

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Are we going to have this question every week now? –  Asaf Karagila Apr 19 '12 at 0:28
    
In the first argument you should explicitly state that the $x_n\in{\text{Ker}}\,T$ converge to $x$. In the second argument, you need to state that the $x_n$ are bounded. –  David Mitra Apr 19 '12 at 0:30
    
@Asaf: Why not vote to close? –  The Chaz 2.0 Apr 19 '12 at 0:34
    
@TheChaz: Because at 3:30 in the morning I feel my judgment in closing a question might be a bit off. –  Asaf Karagila Apr 19 '12 at 0:36
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Yes, it would be fine. I made a mistake in a previous comment. You want to say $|T(x_n)|\rightarrow\infty$ (and with $\Vert x_n\Vert \le 1$ for each $n$). –  David Mitra Apr 19 '12 at 0:48
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marked as duplicate by The Chaz 2.0, Nate Eldredge, azarel, t.b., Asaf Karagila Apr 19 '12 at 7:31

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2 Answers

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It looks good, and is the typical argument, except for two things, one of which is minor:

In the first argument, after you say "let $x_n\in\text{Ker}\,T$", you also need to add "and $x_n\rightarrow x$". The previous sentence does not give that to you.

The more egregious error: In the second argument, you need to state that there is a bounded sequence $(x_n)$ such that $|T(x_n)|\rightarrow\infty$; that is, there is a sequence $(x_n)$ with $\Vert x_n\Vert \le 1$ (say) with $|T(x_n)|\rightarrow \infty$. (And you might want to justify this; though, it's almost automatic from the definition of boundedness.) Without the boundedness of the $x_n$, you would not be guaranteed that the sequence $(x_n')$ converges to $a$.


There is a somewhat quicker route for the forward implication: $\{0\}$ is closed in $\Bbb R$, so, since $T$ is continuous, $T^{-1}(\{0\})$ is closed in $X$.

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Can the closed graph theorem be used? –  Nana May 2 '12 at 14:03
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Clearly if $f$ is continuous then its kernel is closed set. for the converse, assume that $f\neq0$ and that $f^{-1}(\{0\})$ is a closed set. Pick some $e$ in $X$ with $f(e)=1$. Suppose by way of contradiction $||f||=\infty$. Then there exists a sequence $\{x_n\}$ in $X$ with $||x_n||=1$ and $f(x_n)\ge n$ for all $n$. Note that the sequence $\{y_n\}$ defined by $y_n=e-\frac{x_n}{f(x_n)}$, satisfies $y_n\in f^{-1}(\{0\})$ for all $n$ and $y_n\rightarrow e$. Since the set $f^{-1}(\{0\})$ is closed it follows that $e$ must belong to it and consequently $f(e)=0$ which is a contradiction. Thus $f$ is a continuous linear functional.

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