Can I say 1,cosnt,sinnt (n varies over the set of natural numbers) is a basis for the vector space of all functions which are 2π periodic?
I got this doubt because every 2π periodic function can be expressed as an infinite linear combination of these right?
First of all is there a concept of infinite linear combination?Also there are problems regarding convergence right?
Or is it just that I can consider it as a good analogy to linear algebra ideas?
Best Answer
The necessary background or argument for this discussion is found in the functional analysis and in particular, in the $L^{p}$ spaces, i.e., the measurable functions such that $$\int_{X}|f|^{p}d\mu < \infty$$ or, in the real ones $$\int|f(x)|^{p}dx < \infty$$
To study Fourier analysis we will be interested in $L^{2}([-\pi,\pi])$, the fundamental proof is:
this would be a base in a finite dimensional space, as for all $f\in L^{2}([-\pi,\pi])$,
$$ f = \sum_{n\in\mathbb{Z}} \alpha_{n}\phi_{n} $$
It turns out that this sequence can approximate any function of space by linear combinations. Behind this are certain topological notions, such as density and convergence, in addition to the metric notion of completeness. So the knowledge of linear algebra that is generally found in finite-dimensional vector spaces is insufficient.
In case you are interested in the topic, I recommend you review the text of