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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?

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  • $\begingroup$ Is the kernel of $L$ closed? $\endgroup$
    – Narutaka OZAWA
    Commented Apr 2, 2014 at 17:11
  • $\begingroup$ Not necessarily. Another phrasing for the question might be "Does the equivalence of the two topologies follow?" $\endgroup$
    – jmk
    Commented Apr 2, 2014 at 17:31
  • $\begingroup$ Is it clear that such a strongest norm exists? $\endgroup$
    – Nate Eldredge
    Commented Apr 2, 2014 at 20:14

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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 $B(y,\varepsilon)$.

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  • $\begingroup$ 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
    Commented Apr 3, 2014 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
    Commented Apr 3, 2014 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
    Commented Apr 3, 2014 at 13:33
  • $\begingroup$ Your question has a very simple answer: NO. $\endgroup$
    – barcelos
    Commented Apr 3, 2014 at 18:35
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    $\begingroup$ 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.
    Commented Apr 9, 2017 at 6:08

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