L ∞ Spaces of Vector-Valued Functions as Spaces of Continuous Functions

Year 2024, Volume: 19 Issue: 1, 1 - 7, 27.05.2024

Banu Güntürk

https://doi.org/10.29233/sdufeffd.1396580

Abstract

It is proved that for any decomposable perfect measure space (𝑍, 𝒜, 𝜇), the space 𝐿𝜔∗∞ (𝜇, 𝐸*) of essentially bounded weak* measurable functions on 𝑍 to 𝐸* is linearly isometric to the space 𝐶(𝑍,𝐸∗*) of continuous functions on 𝑍 to 𝐸∗*, the latter space is being provided with the supremum norm ‖𝑔‖∞ = sup𝑧∈𝑍‖𝑔(𝑧)‖⁡ where 𝐸∗* stands for the space 𝐸* endowed with its weak* topology.

Keywords

𝐿 ∞ Space, Vector-Valued Functions, Perfect Measure, Hyperstonean Space, Continuous Function Spaces

Thanks

I am grateful to my teacher, the late Prof. Bahaettin Cengiz, who made a valuable contribution to this article.

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