Why does the inner product space $( C[0,1], \| \cdot \|_2$) have an orthonormal family $(e^{\color{red}{2\pi}inx})_{n\in \mathbb{N}}$ ?
An orthonormal family in an inner product space
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functional-analysis
hilbert-spaces
inner-product-space
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1This question makes no sense. What is ∥⋅∥, is it the sup norm on continuous functions? Then what does orthonormality have to do with this? There's no inner product! Are you asking why the family is orthonormal under the usual integral inner product? They are on $[-\pi,\pi]$. For density, check that the family satisfies the criteria for the Stone-Weierstrass theorem for C[0,1]. – 2012-04-06
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0Sorry, It makes sense with the 2-norm – 2012-04-06
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4Your family should probably be $(e^{2\pi i n x})_{n \in \mathbb{Z}}$ – 2012-04-06
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0I think your right, I can see it working in this case, with the normal inner product defined on $\|\cdot\|_2$, it must be a typo in the lecture notes.. – 2012-04-06
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0Nooo! I edited the title but introduced another typo... – 2012-04-07
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0@Tyler: fixed, no worries :) – 2012-04-07
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0This space has many collections of orthonormal elements. That is sort of the point of an inner product space. – 2012-04-07