Formal usage of direct sum in a set of functions

Question

Define $f_i$ as functions indexed by $i\in I$.

Is it ok to say that $\bigoplus_{i\in I}f_i = f=(f_1,f_2,f_3,......)$?

I understand that the direct sum is often used in vector spaces or algebra structures. Is the usage here in functions common? I have never seen this kind of usage before, though.

Answer 1

Direct sums are used on structures, where it is a sum on structures that is also disjoint. Functions are maps not structures. However, if you are speaking about images or kernels of functions of a structure, you can inherit the direct sum (i.e. the image/kernel of a linear map is also a vector space, so you can apply direct sum to those). You see that this definition would cause some inconsistences in the language and should not be used. (There might be cases in category theory, however, were maps are considered as objects/structures. Then this definition would work fine)

I must admit that, that my original answer was not 100% correct. If you have maps between direct sums it is quite common to use the direct sums of maps. For example: Let $V_1,V_2,W$ be vector spaces and $f_1:V_1 \rightarrow W, f_2:V_2 \rightarrow W$ two linear maps. Then it is quite common to write $f_1 \oplus f_2 : V_1 \oplus V_2 \rightarrow W$ for the linear map that is inherited.

Answer 2

Your $f$ is a family of functions, indexed by $i\in I$. I suggest writing $f:=\bigl(f_i\bigr)_{i\in I}\ $. Note that we write $x=(x_1,\ldots,x_n)$ for the vector $\sum_{i=1}^n x_i{\bf e}_i$, and not $x=\oplus_{i=1}^n x_i$.