TITLE: Is it okay to think of functions as of vectors with "uncountable index"
QUESTION [6 upvotes]: 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 ?)

REPLY [2 votes]: 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.