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Sequences:

I like to see a sequence $(u_n) \in \mathbb K ^{\mathbb N}$ as: $$(u_n) = \sum_0^{\infty} u_i\times (e_i)$$

where $u_i$ is $i$-ish term of the sequence and $(e_i)$ are the sequences such as $$(e_0)=(1, 0, 0, ...)$$

$$(e_1)= (0,1,0, ...)$$

Functions:

In complete analogy, I'd like do the same with a function $f\in \mathbb K ^{\mathbb R}$

$$f = \int_{\mathbb R} f(x) e_x$$

where $e_x$ :

$$ \left. \begin{array}{l} \text{if $t=x$ }&1\\ \text{if $t\neq x$}&0 \end{array} \right\} =e_x (t) $$

and $f(x)$ is the value of $f$ at $x$

My question is: is it valid to use the integral sign in this definition? I am used to see this integral symbol with the $dx$ symbol. That's why I ask.

niobium
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  • what is $\mathbb{K}$? – whoisit Sep 24 '22 at 10:04
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    $\mathbb K$ is either $\mathbb R$ or $\mathbb C$ – niobium Sep 24 '22 at 10:06
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    The first problem is that you have not defined with respect to what it is that you are integrating . Integration is done wrt some measure. So what you need is the Dirac measure at $x$ That is take $\delta_{x}(A)=1$ if $A$ contains $x$ and $0$ otherwise. with this you can actually define $f(x)=\int_{\Bbb{R}} f(x),d\delta_{x}$ – Mr.Gandalf Sauron Sep 24 '22 at 10:11
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    I think what you need is an "uncountable sum", such as here: https://math.stackexchange.com/questions/20661/the-sum-of-an-uncountable-number-of-positive-numbers – whoisit Sep 24 '22 at 10:12

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