Vector space

Let $X$ be a nonempty set and let $K$ be a field (in these notes, $K=\mathbb{R}$ or $K=\mathbb{C}$). Suppose we are given:

• an addition map $+:X\times X\to X$, $(x,y)\mapsto x+y$;
• a scalar multiplication map $\cdot:K\times X\to X$, $(\lambda,x)\mapsto \lambda x$.

Then $(X,+,\cdot)$ is a vector space over $K$ if for all $x,y,z\in X$ and all $\alpha,\beta\in K$:

1. (Commutativity) $x+y=y+x$.
2. (Associativity) $(x+y)+z=x+(y+z)$.
3. (Zero) there exists $0\in X$ such that $x+0=x$.
4. (Additive inverses) for each $x\in X$ there exists $-x\in X$ such that $x+(-x)=0$.
5. (Distributivity over vectors) $\alpha(x+y)=\alpha x+\alpha y$.
6. (Distributivity over scalars) $(\alpha+\beta)x=\alpha x+\beta x$.
7. (Compatibility) $\alpha(\beta x)=(\alpha\beta)x$.
8. (Unit) $1x=x$.

Elements of $X$ are called vectors, and elements of $K$ are called scalars.

Vector spaces are the ambient setting for linear combinations , linear subspaces , and linear operators ; adding a norm yields a normed space.

Examples:

• $K^n$ with componentwise addition and scalar multiplication.
• The set of real polynomials on $\mathbb{R}$ with the usual operations.
• The set $F(\Omega)$ of all $K$-valued functions on a nonempty set $\Omega$, with $(f+g)(x)=f(x)+g(x)$ and $(\lambda f)(x)=\lambda f(x)$.