vector space

https://en.wikipedia.org/wiki/Inner_product_space

Not to be confused: vector space vs. vector field

A vector space is an algebraic structure - a set with operations (addition and scalar multiplication) satisfying certain axioms.
A vector field, on the other hand, is a geometric object - a function that assigns vectors to points in space.
You could say a vector field uses vector spaces: at each point in space, the vector it assigns comes from some vector space (usually ${\mathbb{R}}^n$), it is about how vectors vary from point to point in space.

Vector Space

A vectorspace $V$ over a field $F$ is a set equipped with two operations - vector addition and scalar multiplication. For all vectors $u,v,w\in V$ and scalars $\alpha ,\beta \in F$, these operations satisfy:
Vector addition:
Closure under addition: $u+v\in V$
Commutativity: $u+v=v+u$
Associativity: $(u+v)+w=u+(v+w)$
Additive identity: $\exists 0\in V:u+0=u$
Additive inverse: $\exists -u\in V:u+(-u)=0$

Scalar multiplication:
Closure under scaling: $\alpha u\in V$
Scalar distributivity: $\alpha (u+v)=\alpha u+\alpha v$
Vector distributivity: $(\alpha +\beta )u=\alpha u+\beta u$
Scalar associativity: $\alpha (\beta u)=(\alpha \beta )u$
Unit scalar: $1u=u$

Example

The real coordinate space ${\mathbb{R}}^n$ with standard addition and scalar multiplication is the most familiar vector space. Even the trivial space containing just a zero vector forms a vector space.

Any $n$-dimensional vector space over a field $F$ is isomorphic to $F^n$

Meaning there exists a one-to-one correspondence between the elements of the vector space and the $n$-tuples of elements from the field that preserves the vector space operations.
For a basis $\{v_1,v_2,\dots ,v_n\}$ of $V$:

$$\begin{array}{rl}\varphi :V& \to F^n\\ \varphi (a_1{v}_1+a_2{v}_2+\cdots +a_n{v}_n)& =(a_1,a_2,\dots ,a_n)\end{array}$$

For instance, ${\mathbb{R}}^3$ is isomorphic to any 3-dimensional vector space over $\mathbb{R}$, such as the space of all polynomials of degree at most 2 with real coefficients, with the basis $\{1,x,x^2\}$; or the space of all 3D vectors representing physical quantities like velocity or force.

Subspace

A subset $W$ of a vector space $V$ is called a subspace if it is closed under the vector space operations. In other words, adding vectors from $W$ or scaling them by field elements always produces vectors that remain in $W$. The span of vectors $v_1,\dots ,v_k$ consists of all possible linear combinations ${\alpha }_1{v}_1+\cdots +{\alpha }_k{v}_k$ with scalars ${\alpha }_i\in F$. Every span forms a subspace.