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As a follow-up question to what is the norm measuring in function spaces

I just had an idea: How about visualizing function spaces as normal planes. What I have in mind is to have an orthogonal coordinate system where the two axes represent two orthogonal functions (e.g. sin and cos). Different functions, that can be related to these, are then entered into the system according to some distance measure (most easily euclidean): sin would be 1 on the sin-axis, cos 1 on the cos-axis, other functions would be positioned according to the norm and their being composed of these building blocks.

Of course two dimensions are not very much but it would suffice for getting an intuition about these concepts and as a toy model (one could e.g. change the basis functions, the functions that are being dis-assembled, the norm, the boundaries etc.).

Does this make sense? Has anyone ever done this? Are their perhaps even tools to play with?

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This is sort of the idea behind the Fourier transform/series. Like you say, there really isn't much you can do with just a single $\sin$ $\cos$ pair with the same frequency. You'd want to get at least 3 or 4 frequencies so you can see something interesting. – Willie Wong Feb 1 '11 at 15:39
Hum, one idea may be to instead plot on a Kiviat diagram. – Willie Wong Feb 3 '11 at 20:07
up vote 1 down vote accepted

The idea is correct.

If $H$ is a Hilbert Spaces (complete vector spaces with an inner product) sometimes we can construct an orthogonal Schauder Basis, that is:

A sequence $(a_1, \ldots, a_n, \ldots)$ such that $\langle a_i, a_j \rangle = 0 $ if $i \neq j$ and for every $v \in H$ $$v = \sum_{n=1}^{\infty}\alpha_i a_i$$ And $\alpha_i = \langle v , a_i \rangle$

If you are in a function space, then you can think of every $a_i$ as a generator of an axis ($k \alpha$ with $k \in \mathbb{R}$ or $\mathbb{C}$) and the sum $\sum_{n=1}^{\infty}\alpha_i a_i$ as an expression of $v$ as a summation of every component in every axis.

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