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.
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 then 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.