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Dual of essentially compactly supported functions on a hemi-compact Radon space

Let $X$ be a hemicompact Radon space and fix a $\sigma$-finite Radon measure $\mu$ on $X$. Let $L(X_n)$ denote the subspace of $L_{\mu}^1(X)$ of "functions" which are $\mu$-essentially ...
Catologist_who_flies_on_Monday's user avatar
3 votes
1 answer
285 views

Semi-norms on LCS inductive limit of Banach Spaces

Let $(E_n,i_n)_{n\in\mathbb{N}}$ be an direct system of Banach spaces in the category of locally convex spaces (LCSs) with continuous linear maps and let $E_{\infty}$ by their inductive limit. What ...
ABIM's user avatar
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1 answer
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Ultrabornological representation for the space of uniformly continuous functions?

Let $\{\omega_i\}_{i\in I}$ be a non-empty set of increasing (not necessarily strictly) continuous functions preserving $0$. Then, for each $i \in I$ define the space $$ C_{\omega_i}(\mathbb{R}^n,\...
ABIM's user avatar
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1 vote
1 answer
203 views

Continuous function on colimit

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}...
ABIM's user avatar
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5 votes
3 answers
675 views

$L^{\infty}$ as colimit

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 ...
ABIM's user avatar
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1 vote
0 answers
99 views

Gluing together dense subset of Projective Limit in $Ban_1$

Let $(X_n,\pi_n^{m})$ be a countable projective system in the category Ban$_1$ of Banach spaces and short linear maps (is (continuous) linear constructions). Then (co)-completeness of this category ...
ABIM's user avatar
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5 votes
1 answer
279 views

Inductive limit commutes with topological tensor product

Consider $H \left(U \right); U \subset \mathbb{C}$ - space of holomorphic functions with compact-open topology. In this topology, this space is Montel, nuclear and Frechet. I want to take the ...
Ann's user avatar
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2 votes
0 answers
60 views

direct limit in locally convex modules and continuous map

Let we have short exact sequences of LCM over LC algebra $A$ with continuous linear maps $$ 0\to B_j\;{\xrightarrow {\ f_j\ }}\;C_j\;{\xrightarrow {\ g_j\ }}\;D_j\to 0. $$ We can take inductive limit (...
Ann's user avatar
  • 171
5 votes
0 answers
211 views

A strict directed colimit of Hausdorff locally-convex spaces that is not Hausdorff

We work in the category of locally-convex spaces (morphisms are the continuous linear maps). Let $\Lambda$ be a directed set, for every $\lambda \in \Lambda$ let $V_{\lambda}$ be a locally-convex ...
Jonathan Gleason's user avatar