\[ \newcommand{R}{\mathbb{R}} \newcommand{C}{\mathbb{C}} \newcommand{defarrow}{\quad \stackrel{\text{def}}{\Longleftrightarrow} \quad} \]
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.
- \((\R,+,\cdot)\), 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).