Let $L:X\rightarrow Y$ be a surjective linear map from Banach space $(X,||\cdot||_X)$ to vector space $Y$ and denote with $\tau_L$ the final topology on $Y$ induced by $T$.

Is $\tau_L$ equivalent with the topology given by the strongest norm on $Y$ in which $L$ is bounded?

  • $\begingroup$ Is the kernel of $L$ closed? $\endgroup$ – Narutaka OZAWA Apr 2 '14 at 17:11
  • $\begingroup$ Not necessarily. Another phrasing for the question might be "Does the equivalence of the two topologies follow?" $\endgroup$ – jmk Apr 2 '14 at 17:31
  • $\begingroup$ Is it clear that such a strongest norm exists? $\endgroup$ – Nate Eldredge Apr 2 '14 at 20:14

There exists a norm on $Y$ making $L$ bounded if and only if the kernel of $L$ is closed.

However, $\|y\|_Y = \inf\lbrace \|x\|: L(x)=y\rbrace$ is always a semi-norm (that is $\|y\|_Y$ may be $0$ for $y\neq 0$) which induces the final topology $\tau_L=\lbrace A\subseteq Y: L^{-1}(A)$ is open in $X\rbrace$. This is simple: $\tau_L$ is finer than the $\|\cdot\|_Y$-topology since $L$ is continuous w.r.t. $\|\cdot\|_Y$. Conversely, if $y\in A \in \tau_L$ you find $x\in X$ with $L(x)=y$ and since $L^{-1}(A)$ is open there is $\varepsilon>0$ with $T(B(x,\varepsilon)) \subseteq A$ which implies that $A$ contains $B(y,\varepsilon)$.

  • $\begingroup$ So the question is a simple exercise in basic functional analysis. Why there are no downvotes, nor suggestions to close? Is the question such a "research level" one that it is worth answering? $\endgroup$ – TaQ Apr 3 '14 at 10:39
  • $\begingroup$ I agree that this is not research level. Is there any need to close when it is answered and the answer is accepted? $\endgroup$ – Jochen Wengenroth Apr 3 '14 at 12:39
  • $\begingroup$ I don't know whether there is such a need. I am just trying to understand whether there is any consistent attitude in MO to "off-topic" questions. $\endgroup$ – TaQ Apr 3 '14 at 13:33
  • $\begingroup$ Your question has a very simple answer: NO. $\endgroup$ – barcelos Apr 3 '14 at 18:35
  • $\begingroup$ In favour of NOT closing: this is the first discussion you find when you type "final topology" on MO search. This is of "research facilitation" level as many researchers believe (and in many cases rightly) that final topologies are tricky. $\endgroup$ – Duchamp Gérard H. E. Apr 9 '17 at 6:08

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