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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?

Community
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Nana
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  • 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 00:30
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    The second standard argument for $\Leftarrow$ would be to factor $T$ as $X \to X / \ker{T} \to \mathbb{R}$, then the first map is continuous as a quotient map and the second map is continuous as a map between two finite-dimensional spaces ($X/\ker{T}$ is either zero or one dimensional depending on whether $T$ is zero or not). Note also that completeness of $X$ is not used at all, it is only used that it is a normed space. – t.b. Apr 19 '12 at 06:44

2 Answers2

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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 $X$.

David Mitra
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  • Can the closed graph theorem be used? – Nana May 02 '12 at 14:03
  • @David Mitra, I'm studying connected metric spaces, and this question is on my book, they give a suggestion: if the kernel is closed, show that for all "a" in R the set X={x ∈ E; f(x)> a} is open, and for that use the fact that every open ball is connected. However, I don't see how that is helpful. – 0212user Oct 25 '17 at 21:47
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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.

Myshkin
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