A Nachbin family on a Hausdorff completely regular space $X$ is any collection $V$ of upper semi continuous non negative functions with some additional conditions. With such a family is associated a locally convex space $CV(X)$ consisting of all continuous scalar functions $f$ suth that $fv$ is bounded on $X$, for every $v \in V$, with the topology $\tau_V$ given by the semi norms $\|fv\|_{\infty}$, $v \in V$. A weighted topology is any topology given in this way.
Now, a locally convex topology $\tau$ on some linear space $E$ is said to be strict if the strongest locally convex topology on $E$ which coincides with $\tau$ on the bounded sets of $(E, \tau)$ is $\tau$ itself.
When is a weighted topology strict ?
It is clear that if $\tau_V$ is bornological, then it is strict. Also if $\tau_V$ is the topology $\beta$ of Buck, it is strict but not bornological.
This is a very good question.
The start of an answer to this question can be found in
W. H. SUMMERS, DUAL SPACES OF WEIGHTED SPACES, Trans. Amer. Math. Soc. 1970:
http://www.ams.org/journals/tran/1970-151-01/S0002-9947-1970-0270129-5/S0002-9947-1970-0270129-5.pdf
Here, results for a strict topology $\Beta$ (p. 323) and weighted spaced endowed with a weighted topology (p. 324) are considered. Sometimes, it seems that almost everything significant in mathematics was introduced in the 100 year period from roughly 1870 to 1970. See, also,
http://www.ams.org/journals/tran/1969-146-00/S0002-9947-1969-0251521-3/S0002-9947-1969-0251521-3.pdf
Strict, weighted and mixed topologies are introduced in
http://ac.els-cdn.com/0001870876900621/1-s2.0-0001870876900621-main.pdf?_tid=b5afe1c2-9083-11e5-b50a-00000aab0f01&acdnat=1448133286_8cf9ceb0b7f5fe893f6318904b615d20
Weighted topological spaces are considered on RG:
https://www.researchgate.net/profile/Liaqat_Khan2/publication/230774330_On_approximation_in_weighted_spaces_of_continuous_vector-valued_functions/links/0912f504319444621f000000.pdf
and
https://www.researchgate.net/profile/Jasbir_Manhas/publication/259370508_Multiplication_operators_on_weighted_spaces_of_vector-valued_continuous_functions/links/00b4952cf78a966062000000.pdf
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Dear Professor James
Thank you for your interest in my last question. I have had a look at the papers you indicated, but no one of them considers the question in its generality. Those who have considered the strict topology have done it in the space $C_b(X)$ or $C_b(X, E)$ but not in the general case. Notice that a weighted space $CV(X)$ need have no relation with $C_b(X)$. Therefore $\beta$ and the topologies like it cannot de defined in this space.
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