Is the requirement that $u+v$ be in $V$ if $u$ and $v$ are in $V$ a valid axiom for the definition of a vector space? Seems to get skipped

Question

For example, at: https://en.wikipedia.org/wiki/Vector_space

There are 8 axioms that a qualify a set to be a vector space. My professor also gave us 8.

However, a textbook I'm reading states 10 axioms. It has all 8 of the above mentioned axioms, as well as the following two:

If $u$ and $v$ are in $V$, then $u+v$ is also in $V$

If $u$ is in $V$, then $cu$ is in V (where $c$ is an arbritrary scalar)

Why is it that these two axioms aren't explicitly stated in some places? As far as I can't tell it's not that they're included within the 8 existing axioms. I know that these are the requirements for something to qualify as a subspace of a vector space...does that have anything to do with it?

I'd just like to know why these two axioms aren't explicitly stated as they seem very important and don't seem to be indirectly covered by any of the other axioms. Any help is appreciated!

Answer 1

The assertion $(\forall u,v\in V):u+v\in V$ is implicit in the assertion that addition is an operation, since an operation in $V$ is a map from $V\times V$ into $V$.

Answer 2

Note that Wikipedia also lists those two axioms of closure under addition and scalar multiplication. They just don't list them among the axioms. See the below quote from the article you linked:

A vector space over a field $F$ is a set $V$ together with two operations that satisfy the eight axioms listed below.

• The first operation, called vector addition or simply addition $+ : V × V → V$, takes any two vectors $v$ and $w$ and assigns to them a third vector which is commonly written as $v + w$, and called the sum of these two vectors. (Note that the resultant vector is also an element of the set $V$).
• The second operation, called scalar multiplication $· : F × V → V$， takes any scalar $a$ and any vector $v$ and gives another vector $av$. (Similarly, the vector $av$ is an element of the set $V$ ).

Answer 3

In the wikipedia article, addition and multiplication are defined as maps \begin{align} +&:V\times V\to V, \;(v,w)\mapsto v+w\\ \cdot&:F\times V\to V, \;(\lambda,v)\mapsto \lambda\cdot v \end{align} which, by definition imply closure (since the codomain of the maps is $V$).

A question that comes to my mind regarding the 10 axiom treatment: How are addition and scalar multiplication defined that closure is not immediate?