A study on the dual of $C(X)$ with the topology of (strong) uniform convergence on a bornology
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866929648720412672 |
|---|---|
| author | Kumar, Akshay |
| author_facet | Kumar, Akshay |
| contents | This article begins by deriving a measure-theoretic decomposition of continuous linear functionals on $C(X)$, the space of all real-valued continuous functions on a metric space $(X, d)$, equipped with the topology $τ_\mathcal{B}$ of uniform convergence on a bornology $\mathcal{B}$. We characterize the bornologies for which $(C(X), τ_{\mathcal{B}})^*=(C(X), τ_{\mathcal{B}}^s)^*$, where $τ_{\mathcal{B}}^s$ represents the topology of strong uniform convergence on $\mathcal{B}$. Furthermore, we examine the normability of $τ_{ucb}$, the topology of uniform convergence on bounded subsets, on $(C(X), τ_{\mathcal{B}})^*$, and explore its relationship with the operator norm topology. Finally, we derive a topology on measures that shares a connection with $(C(X), τ_{\mathcal{B}})^*$ when endowed with $τ_{ucb}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_19277 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A study on the dual of $C(X)$ with the topology of (strong) uniform convergence on a bornology Kumar, Akshay Functional Analysis General Topology Primary: 54C35, 54C40, 46A16, Secondary: 28A33, 54D70 This article begins by deriving a measure-theoretic decomposition of continuous linear functionals on $C(X)$, the space of all real-valued continuous functions on a metric space $(X, d)$, equipped with the topology $τ_\mathcal{B}$ of uniform convergence on a bornology $\mathcal{B}$. We characterize the bornologies for which $(C(X), τ_{\mathcal{B}})^*=(C(X), τ_{\mathcal{B}}^s)^*$, where $τ_{\mathcal{B}}^s$ represents the topology of strong uniform convergence on $\mathcal{B}$. Furthermore, we examine the normability of $τ_{ucb}$, the topology of uniform convergence on bounded subsets, on $(C(X), τ_{\mathcal{B}})^*$, and explore its relationship with the operator norm topology. Finally, we derive a topology on measures that shares a connection with $(C(X), τ_{\mathcal{B}})^*$ when endowed with $τ_{ucb}$. |
| title | A study on the dual of $C(X)$ with the topology of (strong) uniform convergence on a bornology |
| topic | Functional Analysis General Topology Primary: 54C35, 54C40, 46A16, Secondary: 28A33, 54D70 |
| url | https://arxiv.org/abs/2412.19277 |