I am reading the following wikipedia article on tangent spaces, in particular, this subsection on the definition via the cotangent space. Here is a paraphrasing of the first two sentences of the first paragraph at the given link.
"Consider the ideal $I$ that consists of all smooth functions $f$ from the manifold $M$ to $\mathbb{R}$ that vanish at $x\in M$. Then $I$ and $I^2$ are real vector spaces."
I am bothered by the claim that $I^2$ is a real vector space. It is my understanding that $I$ is the set of all smooth functions from $M$ to $\mathbb{R}$ that vanish at $x$ and that $I^2$ is the set of all functions that can be expressed as the product of two elements of $I$. (If I have gotten anything wrong so far please let me know)
It is not clear to me that $I^2$ is closed under vector sum, as this would require that for any functions $f_1,f_2,f_3,f_4$ in $I$, there must exist functions $g_1,g_2$ such that $f_1f_2+f_3f_4 = g_1g_2$. It is not clear to me that this is the case.
Thanks for the help.
There is usually an abuse of notation when defining sets like $I^2$. In this case, $I^2$ is defined as the space of the functions that look like $$ \sum_{i} f_ig_i $$ where $f_i, g_i \in I$. This is now easily seen to be a real vector space.