# Notational question on Kunneth Formula for de Rham cohomology

I got to learn the Kunneth Formula for de Rham cohomology as following.

$$H^n(X\times Y)=\sum_{n=p+q} H^p(X)\otimes H^q(Y).$$

And I could find same notation from https://www.encyclopediaofmath.org/index.php/K%C3%BCnneth_formula.

Actually, it is quite unfamiliar to use $$\sum$$ for spaces. I think it should be $$\oplus$$ instead of $$\sum$$. It might be stupid question and maybe I am wrong and missing something. I am looking for some clarification for it!

And clear explanation for this would be appreciated! Thanks in advance.

• You can use $\sum_{i\in I} W_i$ for spaces $W_i$ if they are all subspaces of a common vector space $V$. It is the subspace of $V$ consisting of all finite sums of the elements of the $W_i$. Note that the sum decomposition of an element needn’t be unique, so knowing something is a sum is less information than knowing that it is a direct sum. In case of this formula, you know that the sum is direct (and a priori the summands aren’t subspaces of a common space) so using $\bigoplus$ is better. – Eike Schulte Dec 25 '18 at 11:31

Yes, you're right. Indeed we are talking about the direct sum over these spaces. Some people use $$\sum$$ rather than $$\bigoplus$$ in this situation. My advisor actually does the same thing. I suspect that it's a more old-fashioned means of writing the expression. But, anyway, the expression should be read as: $$H^n(X\times Y)=\bigoplus_{p+q=n} H^p(X)\otimes H^q(Y)$$ as you propose.