All Questions
Tagged with limits-and-colimits locally-convex-spaces
10 questions
1
vote
0
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90
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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 ...
3
votes
1
answer
285
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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 ...
- 16.4k
0
votes
1
answer
81
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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
$$
C_{\omega_i}(\mathbb{R}^n,\...
- 5,405
5
votes
3
answers
675
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$L^{\infty}$ as colimit
I recently read a result (in Jarchow's book) that any ultrabornological space can be expressed as a colimit (in the category LCS) of Banach spaces. My question is the following.
Let $\mu$ be a ...
- 4,713
1
vote
1
answer
203
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Continuous function on colimit
Let $X$ be a Banach space and $f:X\rightarrow \mathbb{R}$ be continuous. Suppose that $\{X_n\}_{n \in \mathbb{N}}$ is a strictly nested sequence of sub-Banach spaces, for which $\cup_{n \in \mathbb{N}...
- 5,562
1
vote
0
answers
61
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Refinement: Can $L^1_{loc}$ be represented as colimit?
Let $\mu$ be a $\sigma$-finite measure on a measure space $(\mathbb{R}^d,\Sigma)$. Can $L^1_{\mu,loc}$ be represented as an injective-limit in the category of LCS (locally convex spaces and ...
- 5,405
1
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0
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99
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Gluing together dense subset of Projective Limit in $Ban_1$
Let $(X_n,\pi_n^{m})$ be a countable projective system in the category Ban$_1$ of Banach spaces and short linear maps (is (continuous) linear constructions). Then (co)-completeness of this category ...
- 5,405
5
votes
1
answer
277
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Inductive limit commutes with topological tensor product
Consider $H \left(U \right); U \subset \mathbb{C}$ - space of holomorphic functions with compact-open topology. In this topology, this space is Montel, nuclear and Frechet. I want to take the ...
- 799
2
votes
0
answers
60
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direct limit in locally convex modules and continuous map
Let we have short exact sequences of LCM over LC algebra $A$ with continuous linear maps
$$
0\to B_j\;{\xrightarrow {\ f_j\ }}\;C_j\;{\xrightarrow {\ g_j\ }}\;D_j\to 0.
$$
We can take inductive limit (...
- 4,729
5
votes
0
answers
211
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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 ...
- 16.4k