Mathematical Basics

1.2 Mathematical Basics

The Vector Space
Let \((G,+)\) be a commutative group and \({a, b} \in G\). Let \((K,+,*)\) be a field and \(\alpha, \beta \in K\). \(V\) is a vector space, if

\begin{equation*} \begin{split} \alpha (a + b) & = \alpha a + \alpha b \\ (\alpha + \beta ) a & = \alpha a + \beta a\\ (\alpha \beta ) a & = \alpha ( \beta a)\\ 1 a & = a \end{split} \end{equation*}

(1.2)

Examples:


n-dim. real/complex space	real functions

Linear functions / Linear operators
Let \(V\) be a vector space. \(A\) is a linear function, if:

\begin{equation*} \begin{split} A: V \rightarrow V &, f_1, f_2 \in V, \lambda \in \mathbb{R} \; (\mathbb{C}) \\ A(f_1 + f_2) & = A(f_1) + A(f_2)\\ A(\lambda f_1) & = \lambda A(f_1) \end{split} \end{equation*}

(1.3)

Examples:


matrix \(M\), vectors \(\vec{a}, \vec{b}\)	operator A, function \(f\)

\(M(\vec{a} + \vec{b} ) = M(\vec{a})) + M(\vec{b})\)	a) \(A f = a f\) (multiplication with a constant)

	b) \(A f = x f\) (multiplication with x)

	c) \(A f = \frac{df}{dx}\) (differentiation)

	d) \(A f = \int K(x-y) f(y) dy\) (folding with an integral kernel)

Inner product or scalar product
Let \(V\) be a vector space over the field \(K\) (real or complex space), \(\langle , \rangle : V \times V \rightarrow K\) is called scalar product, if:

\begin{equation*} \begin{split} \langle x,y\rangle & = \langle y,x\rangle ^* \\ \langle x_1 + x_2 , y\rangle & = \langle x_1,y\rangle + \langle x_2,y\rangle \\ \langle x, \alpha y\rangle & = \alpha \langle x,y\rangle \\ \mbox{i.e.} \qquad \langle \alpha x, y\rangle & = \alpha^* \langle x,y\rangle \\ \langle x,x\rangle & \in \mathbb{R}^+ \qquad \mbox{for} \qquad x \neq 0 \end{split} \end{equation*}

(1.4)

Examples:


\(\langle a,b\rangle = \sum_i a_i^* b_i \)	\(\langle f,g\rangle = \int f^*(x) g(x) dx\)
	REMARK: \(f\) and \(g\) should be square integrable

A vector space with a scalar product is call Prehilbert space. A complete Prehilbert space is called Hilbert space. Mathematically: Quantization occurs, because the space of square integrable functions is not complete.
Using the scalar product, some very important properties of vectors can be proofed:
The norm of a vector
is defined by

\begin{equation*} \left|x\right| = \sqrt{\langle x,x\rangle } \end{equation*}

(1.5)

Examples:


\(\left\|a\right\| = \sqrt{\sum_{i=1}^n \left\|a_i\right\|^2}\)	\(\left\|f\right\| = \sqrt{\int \left\|f(x)\right\|^2 dx}\)

Two vectors are orthogonal (perpendicular):

\begin{equation*} \langle x,y\rangle = 0 \end{equation*}

(1.6)

Two vectors are parallel:

\begin{equation*} x = \lambda y \qquad \lambda \in \mathbb{C} \end{equation*}

(1.7)

The Schwarz inequality

\begin{equation*} \left|\left\langle f,g\right\rangle \right| \leq |f| |g| \end{equation*}

(1.8)

is a direct consequence of the definition of the scalar product.
The Triangle inequality

\begin{equation*} |f+g|\leq |f|+|g| \end{equation*}

(1.9)

again is a direct consequence of the definition of the scalar product. It can be transformed into

\begin{equation*} |f-g|\geq |f|-|g| \end{equation*}

(1.10)