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 either $f$ or an infinite-sum of the $f|_{X_n}$..

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

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 $\cup_{n \in \mathbb{N}} X_n$ dense in $X$. 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 (in the category LCS of locally convex spaces and continuous linear maps) 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 either $f$ or an infinite-sum of the $f|_{X_n}$..

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

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.

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 either $f$ or an infinite-sum of the $f|_{X_n}$..

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