$\newcommand{R}{\mathbb{R}} \newcommand{C}{\mathbb{C}} \newcommand{defarrow}{\quad \stackrel{\text{def}}{\Longleftrightarrow} \quad}$

# Vector spaces

## Definition

A real vector space is a set $X$ endowed with an operation called addition, $X \times X \to X, \qquad (x,y) \mapsto x+y,$ an operation called scalar multiplication, $\R \times X \to X, \qquad (\lambda,x) \mapsto \lambda x,$ an element $\mathbf 0 \in X$ called the zero vector, and for each $x \in X$ an additive inverse $-x \in X$, such that for any elements $x,y,z \in X$ and real numbers $\lambda, \mu \in \R$ the following properties hold:

\begin{align*} &\mathrm{(i)} \quad\; &x + \mathbf 0 &= x, \qquad&\text{(additive identity)}&\\ &\mathrm{(ii)} \quad &x + (-x) &= \mathbf 0, \qquad&\text{(additive inverse)}&\\ &\mathrm{(iii)} \qquad& x + y &= y + x, \qquad&\text{(symmetry)}&\\ &\mathrm{(iv)} \qquad &x+(y + z) &= (x+y) + z, \qquad&\text{(associativity)}&\\ \\ &\mathrm{(v)} \qquad &1 x &= x, \qquad&\text{(multiplicative identity)}&\\ &\mathrm{(vi)} \qquad &\lambda(\mu x) &= (\lambda \mu) x, \qquad&\text{(compatibility)}&\\ \\ &\mathrm{(vii)} \qquad &\lambda(x+y) &= \lambda x + \lambda y, \qquad&\text{(distributivity)}&\\ &\mathrm{(viii)} \qquad &(\lambda + \mu)x &= \lambda x + \mu x, \qquad&\text{(distributivity)}&\\ \end{align*} The elements of $X$ are called vectors. If the field of scalars $\R$ is replaced with $\C$ one obtains instead a complex vector space. 1)

N.b. 1 The notion of a vector space and that of a linear space are identitical.
N.b. 2 The elements of a real vector space need not be real-valued. It is the field of scalars that determines whether a vector space is calld real or complex.

Ex.
• $(\R,+,\cdot)$, the set of real numbers $\R$ endowed with the usual addition and multiplication is a real vector space.
• More generally, Euclidean space $\R^n = \{ (x_1, \ldots, x_n ) \colon x_j \in \R \text{ for } j =1,2, \ldots, n. \}$ endowed with componentwise addition $(x_1, \ldots, x_n) + (y_1, \ldots, y_n) = (x_1 + y_1, \ldots, x_n + y_n)$ and componentwise scalar addition $\lambda (x_1, \ldots, x_n) = (\lambda x_1, \ldots, \lambda x_n)$ is a real vector space for any natural number $n \in \mathbb{N}$.
• The set of real-valued continuous functions on an interval $I \subset \R$, $C(I,\R) = \{ f\colon I \to \R \text{ such that } f \text{ is continuous}\}$ is a real vector space with the zero function $f \equiv 0$ as additive identity and $-f$ as additive inverse, when one defines

\begin{align} (f + g)(t) &:= f(t) + g(t),\\ (\lambda f)(t) &:= \lambda f(t),\\ (-f)(t) &:= -f(t). \end{align}

• All three examples above can be turned into complex vector spaces by replacing $\R$ with $\C$, i.e., when both the elements in the space and the field of scalars are replaced. These are the spaces $\C, \quad \C^n \quad\text{ and }\quad C(I,\C).$ The same spaces can also be considered as real vector spaces if the field of scalars is kept to be $\R$. Often, however, spaces that involve complex numbers are regarded as complex vector spaces.
• The essential property of a vector space is linearity: any line $\{ (x,y) = (r\cos(\theta), r\sin(\theta)) \in \R^2 \colon \theta = \theta_0 \}$ is a vector space (addition and scalar mulitplication as in $\R^2$), whereas a closed ball $\{ (x,y) = (r\cos(\theta), r\sin(\theta)) \in \R^2 \colon 0 \leq r \leq \beta \}$ is not (adding or scaling vectors might get one out of the space).
1)
It is possible to define a vector space over any field $\mathbb F$, but we shall not use this.