Let $X$ be a Banach space and $f:X\rightarrow \mathbb{R}$ be continuous.  Suppose that $\{X_n\}_{n \in \mathbb{N}}$ is a strictly nested sequence of sub-Banach spaces, for which $\cup_{n \in \mathbb{N}} X_n =X$.  Then the colimit is an LF-space which is not metrizable. 

Since $f|_{X_n}$ is continuous to $\mathbb{R}$, then by the universal property of the colimit it should extend to $f':X\rightarrow \mathbb{R}$ (where not $X$ is considered with the colimit topology and not its original Banach space topology).  

What is $f'$ (explicitly in tems of $f|_{X_n}$)?  My intuition is that it is eithe $f$ or an infinite-sum of the $f|_{X_n}$..

**Note:** In particular it shouldn't be $f$ because the colimit topology is strictly finer.