I have a confusion about Subspaces and i hope that can be cleared.. Say $E,F,G$ are subspaces of the same vector space. Say we want to show that $(E\cap F)+(E\cap G)$ is a subspace of $E\cap(F+G)$.
My first question is, Does being a subspace necessarily mean being a subset? And if so, to show the mentioned above, do i have to, first, show that $(E\cap F)+(E\cap G) \subset E\cap(F+G)$ ?, i know three conditions have to be satisfied for a set to be a subspace of some vector space, and the first one among them is having the zero vector in it, How do i do that for the one above? I mean i don't know what is the zero vector for this vector space, Is it the same as the zero vector of the vector space of $E,F,G$ ? I was able to show that it is closed under addition and multiplication so that is not my question..
Thanks everyone!
Subset, yes, it is necessary: For $x+y$, $x\in E\cap F$, $y\in E\cap G$, then as $E$ is a vector space and $x,y\in E$, then $x+y\in E$. Also, $x\in F$ and $y\in G$, so $x+y\in F+G$, so we conclude that $x+y\in E\cap(F+G)$.