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In Foundations of Modern Analysis, Jean Dieudonné writes the following:

"This, for instance, enables one to formulate in a reasonable way the theorem on the product of two such series of real numbers, in contrast to the nonsensical so-called "Cauchy multiplication" still taught in some textbooks, and which has no meaning for series other than power series of one variable."

This was written in after he comments about the fact that in absolutely convergent series, the ordering of the terms is completely irrelevant.

It seems to me that after this was written some important theorems about Cauchy multiplication were discovered. Was he right to say that Cauchy multiplication is nonsensical?

EDIT: Due to requests, here is the definition of Cauchy multiplication that I know:

Given $\sum a_n$ and $\sum b_n$, we put $$c_n=\sum_{k=0}^n a_k b_{n-k}$$ and call $\sum c_n$ the Cauchy product of the two given series.

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    You need to explain in mathematical words what you are referring to. – 2017-02-28
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    @user1952009 Could you tell me what should I clarify with mathematical words? I think the context makes it clear that Dieudonné was talking about the concept of Cauchy product of two series of real numbers. – 2017-02-28
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    Everything. Define your question in mathematical words ${}{}{}$ – 2017-02-28
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    @user1952009 To be honest I don't think there is anything to be written in "mathematical words" here. This is not a post about some exercise or some theorem. This is a post about the history of mathematics. I question how valid was Dieudonné's affirmation. – 2017-02-28
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    Come on... Do you understand what means multiplying two series ? If so, can you define it **mathematically** ? – 2017-02-28
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    I do! I've studied Cauchy multiplication in Baby Rudin. Thats why I found Dieudonné afirmation strange. – 2017-02-28
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    @user1952009 Done. – 2017-02-28
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    $\sum_{n=0}^\infty a_n$ and $\sum_{n=0}^\infty b_n$ converges ? Conditionally or absolutely ? The Cauchy product of two power series is $(\sum_{n=0}^\infty a_n z^n)(\sum_{n=0}^\infty b_n z^n) = \sum_{n=0}^\infty c_n z^n$. Why is it natural/justified when $|z| < \min(r,R)$ ? What happens if you add many zeros between the elements of $a_n$, it won't change $\sum_n a_n$ but what will be the effect on $\sum_{n=0}^\infty c_n$ ? What about the product of two Dirichlet series $(\sum_{n=1}^\infty a_n n^{-s})(\sum_{n=1}^\infty b_n n^{-s})$ ? – 2017-02-28

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I don't believe that any “important theorems about Cauchy multiplication” were discovered after Dieudonné wrote this book.

I agree with Dieudonné here. He is talking about expressions of the type$$B\left(\sum_{n=0}^\infty x_n,\sum_{m=0}^\infty y_n\right),\tag{1}$$where $B$ is a continuous bilinear map from $E\times F$ into $G$, where $E$, $F$, and $G$ are Banach spaces and both series are absolutely convergent. At this point of his textbook, he has already defined sums of families of vectors in a Banach space and the he proves (statement 5.5.3) that$$(1)=\sum_{n,m\in\mathbb{Z}^+}B(x_n,y_m).$$How could we express this without the concept of sum of a family of vectors? Well, some authors state that$$(1)=\sum_{n=0}^\infty z_n\text{, with }z_n=\sum_{k=0}^nB(x_k,y_{n-k}),$$calling it “Cauchy multiplication”. Just like Dieudonné, I think that this is rather artificial outside the context of power series.

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    Why did you delete my answer from this question? It is refered to by the accepted answer and it makes no sense to delete it (especially since the other answer considers my answer als valid)! i already discussed this with the previous reviewer who wanted to delete it. https://math.stackexchange.com/questions/1760048/what-is-a-sample-of-a-random-variable/1760069#1760069 – 2017-07-25
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    @Adrian When I have to decide between deleting an answer or not, I do not have access to the other answers; only to the question. And I chose to approve its deletion because it was *not* an answer to the original question, as it has been observed in the comments to it. – 2017-07-25
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    @Adrian This is not the right place to discuss this; it has nothing to do with the question and just adds noise. If you have an issue with this ask a question on [meta](https://math.meta.stackexchange.com/) (or better; post a request [here](https://math.meta.stackexchange.com/questions/19042/requests-for-reopen-undeletion-votes-etc-volume-01-2015-current-versio)) instead. – 2017-07-25
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    @Winther You are right. You wrote what I should have written myself. – 2017-07-25