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In some applied areas that have a little scent of functional analysis (e.g., getting error bound in numerical methods), it is somewhat appealing for me to think of functions $\mathbb R \to \mathbb R$ as of infinite-dimensional vectors $f$ with real "uncountable index" $x \in \mathbb R$, as if $f(x) \approx f_x$, just like addressing the element of countable-index vector could look like $v_3$ or $v_{\frac 3 4}$ (which also seems okay as soon as $\mathbb Q$ it is countable? and vice versa $v \in \mathbb R^3$ can be seen as $v \in \mathbb [0; 3]\cap\mathbb N \to \mathbb R$; and our notation $v_3$ is a certain type of "currying"). Are there any fundamental flaws in this type of thinking that might lead to counter-intuitive errors (something seems valid\invalid if you think of $x$ as just a uncountable index, but is actually invalid\valid; like, something bad happening about completeness or ?)

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  • $\begingroup$ This is a popular point of view in quantum mechanics. Defining $\delta_x$ to be the Dirac delta centered at $x\in\mathbb R$ one obtains a "coordinate expansion" for the function $f$ as follows: $$f(x)=\int_{\mathbb R} f(y)\, \delta_x(y)\, dy$$This is useful as a guiding principle, but has problems in terms of rigorous mathematics. $\endgroup$ Jun 1 '16 at 16:02
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There is nothing wrong about treating functions as being vectors in a space of uncountable dimension. The set of functions $S\rightarrow\mathbb R$ is a vector space under pointwise addition and scalar multiplication for any set $S$, and this is an entirely sensible vector space to think about, so anything that follows from the vector space structure alone will be fine.

However, you ask about notions about completeness, which is a consequence of the structure of a normed vector space, but not of a vector space. This is more problematic, since there's no canonical norm on $\mathbb R\rightarrow\mathbb R$ that I know of. The only common examples I can think of are:

  • The uniform norm on the space of bounded functions $\mathbb R\rightarrow\mathbb R$. This is $\|f\|=\sup_{x\in \mathbb R}|f(x)|$. This space is complete.

  • The $L^p$ norm on an appropriate subspace of the space of functions modulo the equivalence relation $f\sim g$ if $f=g$ almost everywhere. This is $\|f\|_p=\left(\int |f(x)|^p\,dx\right)^{1/p}.$ These spaces are also complete for $1\leq p \leq \infty$.

One can cook up various other norms, but the idea is that the space of functions $\mathbb R\rightarrow\mathbb R$ doesn't have very many properties until you start to give it (or subsets/quotients of it) more structure, such as by including a norm appropriate to the application.

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No, there's nothing particularly wrong with that. It is, for example, perfectly consistent with how sequences are denoted. For exmaple a sequence of real numbers $x_1,x_2,x_3,...$ is, formally, just a function $x : \mathbb{N} \to \mathbb{R}$, and while it is seldom used it is sometimes convenient to use function notation $x(1),x(2),x(3),...$.

But there might be a few ways that this could get you in trouble. These "strategic errors" (if not logical errors) might make it hard for those with training in classical mathematical notation to follow your argument (although if your intended audience understands you, then this might be a non-issue).

  • You might end up confusing the notation with other usages. For instances, sometimes $f_x$ is used for "the partial derivative with respect to $x$".
  • Or, you might end up forgetting that the domain of the function is not a discrete space, as it is with $\mathbb{N}$, and because of this the function has some restrictions on its behaviour, such as continuity or differentiability.
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If you are going to think of basis elements of functions $\delta_{x}$ that are $1$ at $x$ and $0$ otherwise, and if you want to write $f$ as a sum of such functions, then a purely linear algebraic approach won't work because there you have only finite linear combinations of basis elements.

The next thing you might try to do is put a topology on the space in order to allow more general sums. However, even still, you'll have a hard time thinking of a way that you can consider the function to be an uncountable sum. Uncountable sums of orthonormal basis elements are allowed in a Hilbert space, but only if there are no more than a countable number of non-zero coefficients. In general this becomes a problem for normed spaces.

Next you might try some exotic topology that gives you a topological vector space. That takes way too much effort without any real payoff.

You might trying imposing measurability, but it's so easy to lose pointwise properties when you start looking at topologies on Lebesgue spaces.

Or you could start looking at spaces of distributions ... that's a lot of work.

Define your structure carefully, and maybe you'll have something new and interesting.

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  • $\begingroup$ Could you please explain the phrase "it's so easy to lose pointwise properties when you start looking at topologies on Lebesgue spaces" in terms of various convergence modes? $\endgroup$
    – Konstantin
    Mar 2 '17 at 15:10
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    $\begingroup$ @Konstantin : Because such topologies equate functions that are equal a.e. with respect to the underlying measure, pointwise properties are then lost. $\endgroup$ Mar 2 '17 at 18:59

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