Unconditional Convergence of Series in Banach and Hilbert Spaces

Abstract

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In any real or complex vector space X we can always form finite linear combinations cnxn of elements of X. However, we cannot form infinite series or “infinite linear combinations” unless we have some notion of what it means to converge in X. This is because an infinite series xn is, by definition, the limit of the partial sums.Fortunately, we are interested in normed vector spaces. A normed space has a natural notion of convergence, and therefore we can consider infinite series and “infinite linear combinations” in these spaces.