Almost sure linear independence of absolutely continuous Hilbert space-valued random vectors with respect to a special class of Hilbert space probability measures

2025/02/26 by Idrissi, Nizar El, Zoubeir, Hicham · 1 citation
#15A03 #28A20 #46C99 #60B12 #60B20 #FOS: Mathematics #Functional Analysis (math.FA) #Probability (math.PR)

paper · doi:10.48550/arxiv.2502.18908

Abstract

This note examines the implications of randomly selecting vectors from an infinite-dimensional Hilbert space on linear independence, assuming that for all k, the first k vectors follow an absolutely continuous law with respect to a probability measure. It demonstrates that no constraints on the random dimension of their span are necessary, provided that all finite-dimensional vector subspaces are considered negligible with respect to the Hilbert space probability measure.

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• Probability measure annihilating all finite-dimensional subspaces

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