I recently read a result (in Jarchow's book) that any ultrabornological space can be expressed as a colimit (in the category LCS) of Banach spaces. My question is the following. 


Let $\mu$ be a finite measure on $\mathbb{R}$. Is there an uncountable set of $p\_i \in [0,\infty)$ and finite Borel measures $\mu\_i$ on $\mathbb{R}$ for which one can construct an inductive system $\{L^{p\_i}\_{\mu\_i}(\mathbb{R})\}$ of locally convex spaces for which
$$
\injlim L^{p\_i}\_{\mu\_i}(\mathbb{R}) \cong L^{\infty}\_{\mu}(\mathbb{R})?
$$


**Edit:**
Alternatively, I look for a system such that
$
\injlim L^{p\_i}\_{\mu\_i}(\mathbb{R})
$
is dense in $L^{\infty}\_{\mu}(\mathbb{R})$ when its topology is relaxed from the final topology (which need not be metrizable) to the (relative) Banach topology of $L^{\infty}\_{\mu}(\mathbb{R})$.



$L^{\infty}$ as colimit
The answer to this question is YES -- but it is useless! 


In fact, a theorem of Valdivia (which you can find, e.g., as [Theorem 6.5.8](https://books.google.com/books?id=IKrsKlzLcKIC&pg=PA195) in the book *Barrelled Locally Convex Spaces* of Bonet and Perez-Carreras) states that given any infinite-dimensional separable Banach space $F$, one can represent any ultrabornological locally convex space $E$ (this means, that $E$ has some representation as an inductive limit of Banach spaces) which does not carry the finest locally convex topology as an inductive limit of Banach spaces isomorphic to $F$ such that, in addition,the linking maps between the steps are nuclear operators.


This theorem is rather useless because it is too good: *Every* ultrabornological space has such a representation and thus you cannot deduce any nice properties from it. 


