A Universal Approximation Theorem for Neural Networks with Outputs in Locally Convex Spaces
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866910045038444544 |
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| author | Saini, Sachin |
| author_facet | Saini, Sachin |
| contents | In this paper, a universal approximation theorem (UAT) for shallow neural networks whose inputs belong to a topological vector space (TVS) and whose outputs take values in a Hausdorff locally convex TVS is established. The networks are constructed using scalar activation functions applied to continuous linear functionals of the input, while the output coefficients lie in the target space. It is shown that this class of neural networks is dense in the space of continuous mappings from a compact subset of the input space into the target space with respect to the topology of uniform convergence induced by the defining seminorms. The result extends existing scalar-valued approximation theorems on TVS and includes Banach and Hilbert-valued approximation as special cases. Several corollaries and examples are provided, illustrating applications to operator approximation between function spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_07242 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Universal Approximation Theorem for Neural Networks with Outputs in Locally Convex Spaces Saini, Sachin Functional Analysis 41A30\sep 41A65\sep 46A20\sep 46A22\sep 68T05 In this paper, a universal approximation theorem (UAT) for shallow neural networks whose inputs belong to a topological vector space (TVS) and whose outputs take values in a Hausdorff locally convex TVS is established. The networks are constructed using scalar activation functions applied to continuous linear functionals of the input, while the output coefficients lie in the target space. It is shown that this class of neural networks is dense in the space of continuous mappings from a compact subset of the input space into the target space with respect to the topology of uniform convergence induced by the defining seminorms. The result extends existing scalar-valued approximation theorems on TVS and includes Banach and Hilbert-valued approximation as special cases. Several corollaries and examples are provided, illustrating applications to operator approximation between function spaces. |
| title | A Universal Approximation Theorem for Neural Networks with Outputs in Locally Convex Spaces |
| topic | Functional Analysis 41A30\sep 41A65\sep 46A20\sep 46A22\sep 68T05 |
| url | https://arxiv.org/abs/2603.07242 |