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arXiv · 2502.18908

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

Nizar El Idrissi·Hicham Zoubeir
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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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Nizar El Idrissi, Hicham Zoubeir. 2025-02-26. Almost sure linear independence of absolutely continuous Hilbert space-valued random vectors with respect to a special class of Hilbert space probability measures. https://arxiv.org/abs/2502.18908

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