# Question about notation on the subject of Vector Spaces

I'm reading a book from dover publications called Introduction to LINEAR ALGEBRA by Marvin Marcus and Henryk Minc. It page 6 it says:

If X is any subset of V (a vector space) then $\langle X \rangle$ will denote the totality of linear combinations of vectors in X. The set $\langle X \rangle$ is called the space spanned by X. Each element of $\langle X \rangle$ is contained in V. We designate that by $\langle X \rangle \subset V$.

My problem arises in the last statement. I was expecting $\langle X \rangle \subseteq V$ instead of $\langle X \rangle \subset V$.

Now I know that if we had two sets A and B and the relationship: $$A \subset B$$ holds (which means that A is a proper subset of B), that means that every member of A is also a member of B ( $A \subseteq B$) AND $A \neq B$.

Some further thoughts:

Suppose $V_3(\mathbb R)$ be a vector space. The set $I = \{ (1,0,0) , (0,1,0), (0,0,1) \} = \{ \hat i , \hat j , \hat k \}$ is clearly a proper subset of our vector space. So $I \subset V_3(\mathbb R)$. However we know that I is a basis for $V_3(\mathbb R)$. And therefore the totality of linear combination of the vectors in I gives you $V_3(\mathbb R)$. Hence: $$\langle I \rangle = V_3(\mathbb R)$$ which contracdicts the definition from the book. Where am I wrong ? Thanks in Advance!

• Some (most?) books don't distinguish "$\subset$" from "$\subseteq$", check out your book notation in the Index of Symbols section – francescop21 Jul 12 '18 at 13:52
• See subset : symbols. – Mauro ALLEGRANZA Jul 12 '18 at 13:53
• Oh really? hahaha that was easy! Thank you very much ! ! – Andreas Mastronikolis Jul 12 '18 at 13:56

• If $A$ is a subset of $B$ which can possibly be equal to $B$, then we write $A \subset B$. However, if $A$ is a subset of $B$ which cannot be equal to $B$, then we write $A \subsetneq B$, with the bar on the bottom to indicate that $A$ is a proper subset of $B$. This is the more common notation.
• The other , which is what you must be used to, is that $A \subset B$ means that $A$ cannot equal $B$, while $A \subseteq B$ means $A$ can equal $B$. This is not as common.
Of course, you are right to realize that there can be equality. However, rather than letting $X$ be a basis for $V$, you could let $X = V$ itself : that itself would show that there is equality in the containment relation.