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I am having some difficulty in understanding some concepts regarding Hilbert spaces. I am learning wavelet theory (with regards to signal processing) and am reading up some basics on signal decomposition. I'm pursuing a text called 'Wavelets and Subband Coding' by Vetterli. I came across the following definitions:

Consider s subset E of a vector space. "If every Cauchy Sequence of vectors {xn} in E converges to a vector in E, then E is called complete.

"A complete inner product space is called a Hilbert space."

And then under the topic Orthonormal Bases, I read: "For a set of vectors S = {xi} to be an orthonormal basis, we first have to check that the set of vectors S is orthonormal and then that it is complete, that is, that every vector from the space to be represented can be expressed as a linear combination of the vectors from S."

This seems to imply that completeness means that S spans the space. Is this the correct interpretation? Does completeness imply spanning or vice versa? How does this relate with the definition of completeness regarding convergence of Cauchy sequences?

A simple explanation would be much appreciated. I know only the basics of linear algebra and the concept of cauchy sequences, Hilbert spaces etc. are very new to me. I searched online, but whatever explanations I see seemed to be full of abstract terms.

EDIT: Orthonormal Basis for Hilbert Spaces seems to discuss a similar topic, but I am struggling to understand it.

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The issue is in the sentence: then that it is complete, that is, that every vector from the space to be represented can be expressed as a linear combination of the vectors from $S$.

A linear combination is supposed to be a finite sum. In a Hilbert space, any vector $x$ can be represented as a potentially infinite sum of the vectors $x_i$ of the Hilbert basis.

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