Bases of non-negative functions in Hilbert spaces and free Banach lattices

The HADES seminar on Tuesday, February 18th will be given by Mitchell Taylor in Evans 740 from 3:40 to 5 pm.

Speaker: Mitchell Taylor, Berkeley

Abstract:  A basis of a Banach space $X$ is a sequence $(x_k)$ in $X$ such that for every $x\in X$ there is a unique sequence of scalars $(a_k)$ such that $x=\sum_{k=1}^\infty a_kx_k$. Examples of bases in $L_2([0,1])$ include the Haar, Walsh, and trigonometric bases. A question arising independently in Engineering and Stochastic PDE is whether $L_p([0,1])$ admits a basis with each of the $x_k$ being a non-negative function. It is a theorem of Bill Johnson and Gideon Schechtman that $L_1$ admits such a basis, and that any non-negative basis in $L_p$ must necessarily be conditional, i.e., it will fail to be a basis if the $(x_k)$ are permuted. In this talk I will give a construction of a non-negative basis in $L_2$, and at the end will discuss non-negative bases in general; in particular, the connection to free Banach lattices.

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