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Isn't this the opposite way ? Vectors are functions whose input space are discrete dimensions. Let's not pretend going from natural numbers to real is "simple",

Isn't this the opposite way ? Vectors are functions whose input space are discrete dimensions. Let's not pretend going from natural numbers to real is "simple", reals numbers are a fascinating non-obvious math discovery. And also the passage from a few numbers to all natural numbers (aleph0) is non obvious. So basically we have two alephs passages to transforms N-D vectors as functions over reals.

xeonmc 1y ago
Vectors are not necessarily discrete-domained. Anything that satisfies the vector space properties is a vector.
- ttoinou 1y ago
  I agree but I'm operating under the assumption of the article Conceptualizing functions as infinite-dimensional vectors lets us apply the tools of linear algebra to a vast landscape of new problems
  - layer8 1y ago
    Linear algebra isn’t limited to discrete-dimensional vector spaces. Or what do you mean?
    - ttoinou 1y ago
      See my other comment sibling. And he's starting from the assumption vectors are finite (cf. the article)
      - Sharlin 1y ago
        He does not assume anything! Any assumption is in your head only. Of course he starts from the specific type of vector spaces that's the most familiar to readers. But then he shows that there's nothing that requires a vector space to have a finite, or even countably infinite, dimension. What matters are the axioms.
        ttoinou 1y ago
        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. So the goal is to impress the reader by letting him believe we will easily apply our easy linear algebra to real -> real functions. But we can't.
        bmacho 1y ago
        > So the goal is to impress the reader by letting him believe we will easily apply our easy linear algebra to real -> real functions. But we can't. 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.
        ttoinou 1y ago
        What intuition can you gain ? This applies to any field, not just functions. You're gaining intuition for a more general concept and not what makes real->real functions different from others maths object
  - MITSardine 1y ago
    That's not really the assumption of the article, that's the assumption the article makes of the reader's knowledge of vectors as they begin reading. The article is trying to make the point that functions are also vectors, strictly speaking, and give some intuition as to what that means. Basically 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.
- deleted 1y ago
  [deleted]
gizmo686 1y ago
Vectors are an abstract notion. If you have two sets and two operations that satisfy the definition of a vector space, then you have a vector space; and we refer to elements of the vector set as "vectors" within that vector space. 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.
- ttoinou 1y ago
  I agree but we're using functions for different things here. Yes some specific families of functions can be treated as vector spaces. In this article it seems like the author is pretending to take all real->real functions and treating them as if they are a vector space, whatever the content of the functions, quote : we’ve built a vector space of functions and later he admits it is impossible 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: They are uncountable because they are aleph1
  - 998244353 1y ago
    The set of all real->real functions is still a vector space. 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.
    - ttoinou 1y ago
      I'd love to read more about that, he's not talking about that at all in this article though
bmacho 1y ago
> Vectors are functions whose input space are discrete dimensions. 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.

xeonmc 1y ago

Vectors are not necessarily discrete-domained. Anything that satisfies the vector space properties is a vector.

ttoinou 1y ago
I agree but I'm operating under the assumption of the article Conceptualizing functions as infinite-dimensional vectors lets us apply the tools of linear algebra to a vast landscape of new problems
- layer8 1y ago
  Linear algebra isn’t limited to discrete-dimensional vector spaces. Or what do you mean?
  - ttoinou 1y ago
    See my other comment sibling. And he's starting from the assumption vectors are finite (cf. the article)
    - Sharlin 1y ago
      He does not assume anything! Any assumption is in your head only. Of course he starts from the specific type of vector spaces that's the most familiar to readers. But then he shows that there's nothing that requires a vector space to have a finite, or even countably infinite, dimension. What matters are the axioms.
      - ttoinou 1y ago
        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. So the goal is to impress the reader by letting him believe we will easily apply our easy linear algebra to real -> real functions. But we can't.
        bmacho 1y ago
        > So the goal is to impress the reader by letting him believe we will easily apply our easy linear algebra to real -> real functions. But we can't. 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.
        ttoinou 1y ago
        What intuition can you gain ? This applies to any field, not just functions. You're gaining intuition for a more general concept and not what makes real->real functions different from others maths object
- MITSardine 1y ago
  That's not really the assumption of the article, that's the assumption the article makes of the reader's knowledge of vectors as they begin reading. The article is trying to make the point that functions are also vectors, strictly speaking, and give some intuition as to what that means. Basically 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.
deleted 1y ago
[deleted]

gizmo686 1y ago

Vectors are an abstract notion. If you have two sets and two operations that satisfy the definition of a vector space, then you have a vector space; and we refer to elements of the vector set as "vectors" within that vector space. 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.

ttoinou 1y ago
I agree but we're using functions for different things here. Yes some specific families of functions can be treated as vector spaces. In this article it seems like the author is pretending to take all real->real functions and treating them as if they are a vector space, whatever the content of the functions, quote : we’ve built a vector space of functions and later he admits it is impossible 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: They are uncountable because they are aleph1
- 998244353 1y ago
  The set of all real->real functions is still a vector space. 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.
  - ttoinou 1y ago
    I'd love to read more about that, he's not talking about that at all in this article though

bmacho 1y ago

> Vectors are functions whose input space are discrete dimensions. 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.