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This is from Kreyszig's functional analysis text, chapter 3.6 #2.

The backward direction is easy and just the Gram Schmidt process on any basis. I am having some trouble on the forward direction. So we start with a totally orthonormal set M such that the closure of the span is the entire space X (by definition of total orthonormal). I need to show that in fact, any element can be expressed as the span so M is in fact the basis. For any any element x in X, we can approximate it as close as we want by linear combinations of M, but I'm not sure how to proceed from here. The Parseval relation was introduced in this chapter, but I don't see any way to use it.

Bill
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1 Answers1

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Let $\{e_1, \ldots, e_n\}$ be your finite orthonormal set such that $$\overline{\operatorname{span}\{e_1,\ldots, e_n\}} = X.$$ But a finite-dimensional subspace of $X$ is always closed and hence equal to its closure, so $$\operatorname{span}\{e_1,\ldots, e_n\} = \overline{\operatorname{span}\{e_1,\ldots, e_n\}} = X.$$ Hence $\{e_1,\ldots, e_n\}$ is an algebraic basis for $X$.

mechanodroid
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