A vector space $\small \mathbb V$ is the association of a set of vectors and a set of scalars (or a field), e.g:
- $\small \mathbb R^3$ for the vectors,
- $\small \mathbb R$ for the scalars,
with addition and scalar multiplication satisfying the vector space axioms.
Indeed, for any vector $v$ of this example: $v \in \mathbb R^3$. However it's common to see statements like $v \in \mathbb V$. E.g:
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called vectors, may be added together and multiplied ("scaled") by numbers called scalars.
My questions. Is it correct to say:
- a vector space is a set?
- vectors are elements of the vector space?
Best Answer
Saying that a vector space is a set whose elements can be added etc. can be seen as a remnant of an "Eulerian" worldview. In this view, objects such as numbers have an inherent concept of arithmetic operations such as addition and multiplication. $2 + 3 = 5$ no matter whether you are considering the integers or the complex numbers, and they are the same $2$, $3$ and $5$. In this worldview, being a vector space is a property. A set is either a vector space, or it's not.
Modern mathematics tend to take an alternative worldview. Here, the elements of a set has no internal structure, $2 \in \mathbb R$ is merely a label for an element, and switching labels around doesn't affect mathematics whatsoever. To meaningfully talk about addition on $\mathbb R$, you need to equip it with additional structure, i.e. the structure of a function $\textsf{add} : \mathbb R^2 \to \mathbb R$. In this worldview, a set can become a vector space in multiple ways.
So saying a vector space $\mathcal V$ "is" a set doesn't make sense, in the say way that saying a two dimensional point $(x,y)$ "is" the real number $x$ doesn't make sense. Since there may be different points $(x,y_1), (x, y_2)$ that have the same $x$. Similarly, an inner product space $\mathcal V$ isn't a vector space, and a metric space isn't a topological space. To contrast this, it does make sense to say a finite-dimensional vector space $\mathcal V$ is a vector space, since being finite dimensional is a property. And a metrizable space is a topological space. This distinction of stuff, structure and property has already occurred to many mathematicians, and it is articulated and promoted by John Baez, Michael Shulman and others.
With this in mind, it's easy to see why saying "a vector is an element of a vector space" is a correct but less-than-useful statement. Given a mathematical object $v$, it is no inherent property of $v$ that $v$ belongs to some vector space. You can construct any set $V$ with $v$ in it, and $v$ will become a vector in $V$. What's more useful is saying "$v$ can be viewed as a vector in the vector space $V$". This is telling the reader to think about $v$ in a different and possibly fruitful way.
Of course, usually when people say "Let $V$ be a vector space", what they usually mean is "Let $(V, +, \times, \dots)$ be a vector space", and the letter $V$ actually refers to the set instead of the vector space. So notations such as $x \in V$ or $W \subset V$ makes sense because they are really just notations of sets. But this is just a matter of trivial notation fiddling, and doesn't bear much value unless there is danger of confusion.