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There is a section titled Inner products and norms in my lecture notes on the Fourier series that has been confusing me (the course I'm taking is Higher Several Variable Calculus). The notes go something like this:

  1. They let $V$ be a vector space and define what an inner product on $V$ is.
  2. They then show that $\mathbb{R}^n$, a vector space, "admits" the inner product $$\langle \mathbf{u}, \mathbf{v} \rangle = \sum_{i=1}^n u_i v_i$$ for $\mathbf{u} = (u_1, ..., u_n), \mathbf{v} = (v_1, ..., v_n) \in \mathbb{R}^n$.
  3. They show that the vector space $C[a, b]$ consisting of all continuous functions defined on the interval $[a, b]$ also admits the inner product $$\langle f, g \rangle = \int_a^b f(x)g(x) \ dx$$ for $f, g \in C[a, b]$.
  4. They mention that the property of orthogonality still applies to those functions in $C[a, b]$.
  5. They then do something similar with norms on vector spaces.

Now I have practically no knowledge of abstract algebra besides a rudimentary understanding of vector spaces. Showing that $\mathbb{R}^n$ and $C[a, b]$ admit inner products and norms is all well and good — my question is: what's the point? Is it not sufficient just to show that $C[a, b]$ is a vector space to do things like perform vector decomposition and determine Fourier coefficients?

user26857
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Jeremy Lindsay
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    Here is an example of a post where the notion of an inner product provides a helpful perspective. In more complicated vector spaces such as $C[a,b]$, the existence of an inner product becomes that much more important. – Ben Grossmann Apr 14 '20 at 05:23
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    Ultimately, the punch line in the context of Fourier series (of continuous time functions) is that coefficients can be extracted by taking an inner product, since the sinusoids with which we build periodic functions form an orthonormal basis. In vector spaces without an inner product, finding the coefficients associated with a particular basis requires solving a linear system of equations (over an arbitrarily high number of or possibly infinite set of variables) – Ben Grossmann Apr 14 '20 at 05:26

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It is not sufficient, since you need the Fourier series to converge you need a definition of norm. To find the Fourier components you project your function (using the inner product) onto the basis of sines and cosines.

Basel J.
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