Abstract
\(C_p(X)\) denotes the space of continuous real-valued functions on a Tychonoff space X with the topology of pointwise convergence. A locally convex space (lcs) E with the weak topology is denoted by \(E_{w}\). First, we show that there is no a sequentially continuous linear surjection \(T: C_p(X)\rightarrow E_w\), if E is a lcs with a fundamental sequence of bounded sets. Second, we prove that if there exists a sequentially continuous linear map from \(C_p(X)\) onto \(E_w\) for some infinite-dimensional metrizable lcs E, then the completion of E is isomorphic to the countable power of the real line \({\mathbb {R}}^{\omega }\). Illustrating examples are provided.
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Acknowledgements
Jerzy Ka̧kol is supported by the GAČR project 20-22230L and RVO: 67985840.
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Ka̧kol, J., Leiderman, A. Continuous linear images of spaces \(C_p(X)\) with the weak topology. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 116, 129 (2022). https://doi.org/10.1007/s13398-022-01250-4
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DOI: https://doi.org/10.1007/s13398-022-01250-4