What is a closed operation in a vector space? I don't see any difference between a closed operation in some vector Space R$^n$ and the open operation.
What I mean by the closed operation is addition sign that looks like $\oplus$
Also, is a subspace an element of a vector space R$^n$ or no? Like I must first assume that there is a space R$^n$ which is closed first.
Thank you for the help. (Thank you everybody) ありがとう 皆さん
The $\oplus$ and $\otimes$ symbols are commonly used when you have something like our standard notions of addition and multiplication, but they may be a bit different. For example we might define $x\oplus y=x+y+1$, and $c\otimes x=(c-1)x$. This is a useful notation in terms of a vector space because we want to define two operations on vectors which work similarly to addition and multiplication in that we want to be able to apply associativity, commutivity, and distributivity, but the operations may not actually be the usual ones.
Additionally, you can't quite say that a subspace is an element of a vector space. By definition the vector space contains vectors, not spaces. However, you can say that a subspace is a subset of a vector space, but this is getting a little silly. All a subspace means is that it is a vector space contained in some larger vector space (and both spaces are using the same notion of addition and scalar multiplication).