Do the eight axioms of vector space imply closure?

This post is similar to my question but I do not quite understand the explanation.

Usually, when we try to prove whether a set is a vector space, we will check closure on summation and multiplication, as well as the axioms of vector space.

But, why books like Linear Algebra Done Wrong, when they talk about vector space, they only talk about the 8 axioms?

I am wondering if it is somewhere inside definitions or from the 8 axioms.

• Actually, there is only one axiom for vector spaces. "A vectors space over a field $k$ is an abelian group together with an action of $k$" ;) Aug 18, 2019 at 13:53
• @HagenvonEitzen: How do you define action of $k$? Aug 18, 2019 at 14:06
• Linear Algebra Done Wrong? That's where you don't worry about axioms. Axioms are for people who are too afraid to make up the rules as they go. Sep 24, 2020 at 3:06

2 Answers

Treil says "A vector space $$V$$ is a collection of objects, called vectors..., along with two operations, addition of vectors and multiplication by a number (scalar), such that ...".

He is using operation in sense of functions $${+}: V \times V\to V$$ and $$\cdot: {\mathbb F} \times V \to V$$. In particular he is assuming closure as part of the defintion of an operation. When you need to check (for instance) that a subset $$W$$ of a vector space $$V$$ is itself a vector space (under the "same" operations), that means you need to check that $$+$$ and $$\cdot$$ are in fact operations on $$W$$. This requires that you check $$W$$ is closed under addition and scalar multiplication.

• for the part $+: V \times V \mapsto V$, I am not familiar with the notation. Can you explain a bit more? Why $+$ is a function from $V \times V$ to $V$?
– JOHN
Aug 18, 2019 at 14:12
• Addition is a function that takes an ordered pair as an input, and gives a single element as output. In real numbers, for example, $+(3,5)=8$. Aug 18, 2019 at 14:28
• @John: You may want to read about binary operation for context. Aug 18, 2019 at 14:29

The OP should note that we usually denote a function from $$V \times V$$ to $$V$$ by a letter, say $$f$$, and write

$$f: V \times V \to V$$

and denote the image of $$(v,w) \in V \times V$$ under $$f$$, an element in $$V$$, by $$f(v,w)$$.

Now if the operation is addition, $$+$$, of vectors, we find it both convenient and illuminating to use $$\vec v + \vec w$$ as opposed to $$+(v,w)$$.

Again, we might want to use some letter $$f$$ to denote a function from $$\Bbb R \times V$$ to $$V$$, so that if $$\alpha \in \Bbb R$$ and $$v \in V$$ the image under $$f$$ of $$(\alpha,v)$$ is denoted by $$f(\alpha,v)$$.

But if the operation is scalar multiplication of a vector, $$\cdot$$, we find it both convenient and illuminating to use $$\alpha \cdot \vec v$$ as opposed to

$$\cdot(\alpha,v)$$

In fact, when multiplying by a scalar, the multiplication symbol is usually dropped when no ambiguity can occur, and juxtaposition of the scalar on the left of the vector is an appreciated notation/convention,

$$\alpha \vec v$$

Note that in 1. Vector spaces of the book the author uses juxtaposition for scalar multiplication; see also the first example in 1.1. Examples.