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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

Kagamine Len
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    Could you give your definitions of closed operation and open operation? I don't think this is standard terminology – Peter Woolfitt Jan 29 '15 at 22:56
  • Sorry, the operations of the adding and scalar multiplication is the operations on the space. My question is, is there a different in using + and * and inscribing a + and * sign in two circles: So what I mean is the: vector boldubold + boldv = boldvbold + boldubold. (this is without the inscription) and vector boldubold + boldv = boldvbold + boldubold. (this is with the inscription) – Kagamine Len Jan 29 '15 at 22:57
  • I don't understand the question. How do you define closed, open and $\oplus$? – copper.hat Jan 29 '15 at 23:08
  • yeah sorry, I don't get some of these definitions. Thank you Peter Woolfitt. I didn't understand the definition of the direct sum of the vector space so I didn't know how to phrase it. Thank you for the help. – Kagamine Len Jan 29 '15 at 23:18

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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).

Peter Woolfitt
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