Conceptualizing functions as infinite-dimensional vectors lets us apply the tools of linear algebra to a vast landscape of new problemsBasically it's an introduction to "functional analysis", the field of math that looks at functions like vectors. The term is in there sneakily, but this is really what the article is all about.
I had classes on this subject and we were encouraged to build upon linear algebra intuition (much like the author does), with some caveats (such as a bounded closed set is not necessarily compact in infinite dimension). Things break down, as the author hints at when they mention they didn't prove the Laplacian to be self-adjoint but only symmetric (this has to do with the operator's domain, a concern that doesn't exist for linear operators on finite dimensional spaces)... but it's still a very convenient way to think and talk about things, conceptually.
And he's starting from the assumption vectors are finite (cf. the article)
You may recall that this representation is only one example of an abstract vector space. There are many other types of vectors, such as lists of complex numbers, graph cycles, and even magic squares.
However, all of these vector spaces have one thing in common: a finite number of dimensions. That is, each kind of vector can be represented as a collection of N N numbers, though the definition of “number” varies.
When you only have a finite number of functions (and the space they span), then you can apply your finite dimensional linear algebra, since n dimensional vector spaces over a field are isomorphic. You absolutely can gain intuition for real functions from arrows on a paper type vectors.
E.g. the article mentions the Cauchy-Schwartz inequalities for functions, that's something you can intuit when you imagine our functions being little arrows on a paper.
The observation here is that set of real value functions, combined with the set of real numbers, and the natural notion of function addition and multiplication by a real number satisfies the definition of a vector space. As a result all the results of linear algebra can be applied to real valued functions.
It is true that any vector space is isomorphic to a vector space whose vectors are functions. Linear algebra does make a lot of usage of that result, but it is different from what the article is discussing.
we’ve built a vector space of functions
Ideally, we could express an arbitrary function f as a linear combination of these basis functions. However, there are uncountably many of them—and we can’t simply write down a sum over the reals. Still, considering their linear combination is illustrative:
However, the particular vector space in question (functions from R to R) does have a basis, which the author describes. That basis is not as useful as a basis typically is for finite dimensional (or even countably unfitine dimensional) vector spaces, but it still exists.
> However, the particular vector space in question (functions from R to R) does have a basis, which the author describes.
No, there is no known constructible basis for R -> R functions.
(Imagine/remember what it feels like when children first learn that the integers aren't the only number, there are also fractions, then irrationals, then complex numbers...this is a very similar situation).
With that in mind, you may want to reread the text and pay attention to the definitions he is using, and not assume that your definitions are the whole story.
(Imagine/remember what it feels like when children first learn that the integers aren't the only number, there are also fractions, then irrationals, then complex numbers...this is a very similar situation).
This vector space also has a basis (even if it is not as useful): there is a (uncountably infinite) subset of real->real functions such that every function can be expressed as a linear combination of a finite number of these basis functions, in exactly one way.
There isn't a clean way to write down this basis, though, as you need to use Zorn's lemma or equivalent to construct it.
I think what I may be asking is “Does the complex Fourier transform make a Hilbert space?” but I might be wrong both about that and about that being the right question.
Another example is the eigenvectors of linear operators like the Laplacian. Recall how, in finite dimension, the eigenvectors of a full rank operator (matrix) form an orthonormal basis of the vector space. There is a similar notion in infinite dimension. I can't find an English page that covers this very well, but there's a couple of paragraphs in the Spectral Theorem page (https://en.wikipedia.org/wiki/Spectral_theorem#Unbounded_sel... ). The article linked here also touches on this.
Regarding your last sentence, one thing to note is that having a basis is not what makes you a Hilbert space, but rather having an inner product! In fact, to get the Fourier coefficients, you need to use that inner product.
You can represent any function f: [-pi, pi] -> R as an infinite sum
f(x) = sum_(k = 0 to infinity) (a_k sin(kx) + b_k cos(kx))
This is very useful, but the functions sin(x), sin(2x), ... , cos(x), cos(2x), ... don't constitute a basis in the formal sense I mentioned above as you need an infinite sum to represent most functions. It is still often called a basis though.
f(x) = sum n=-infty to +infty C_n e^{i n x}
sin x = (e^(ix) - e^(-ix))/2i
cos x = (e^(ix) + e^(-ix))/2
The convergence criteria for Fourier series vary depending on how strongly you need convergence but I think basically if a function is differentiable on the interval you care about then the Fourier series provably converges on that interval and otherwise if it has jump discontinuities and that sort of thing, then depending on whether it is square-integrable or a bunch of other properties) you can prove weaker forms of convergence (absolute, pointwise etc).
To address your comment I don’t see why an infinite sum prevents something being a basis. In fact I would specifically say that can’t be true because then there would never be a basis for any infinite-dimensional space- any time you want to take an inner product in such a space you need an infinite sum, and you need such an inner product to construct the basis. A sibling comment pointed me in the direction of a Hilbert basis, which seems to be what I was thinking of.
This is _also_ true. But the fact that finite dimensional real vectors can be viewed as a special case of set-theoretic functions or as a special case of real functions on a discrete finite space is probably less useful than the opposite: the set of real functions have a vector-space structure, and you can use all the neat theorems about (finite or infinite dimensional) vector spaces.
Well, at least the article is about the latter direction.