Math: The Hilbert Space

We start with a few mathematical terms that are necessary for the definition of a Hilbert space.

Def.: A norm $\Vert..\Vert$ is a mapping of a complex vector space $V$ into the real numbers $R_+$, such that for elements $\psi,\phi \in V$

$$\Vert\psi\Vert \ge 0,\quad \Vert\psi\Vert =0 \leftrightarrow \psi=0$$

$$\Vert c\psi\Vert = \vert c\vert \Vert\psi\Vert,\quad c\in C$$

$$\Vert\psi+\phi\Vert \le \Vert\psi\Vert+\Vert\phi\Vert$$ (166)

Def.: A scalar product is a mapping of a pair of vectors $\psi, \phi$ to a complex number $\langle \psi\vert \phi\rangle$ such that for arbitrary $\psi,\phi,\chi\in V$

$$\langle \psi\vert \psi\rangle \ge$$ 0

$$\langle \psi\vert c\phi\rangle = c \langle \psi\vert \phi\rangle , \quad c \in C$$

$$\langle \psi +\phi \vert \chi \rangle = \langle \psi\vert \chi\rangle + \langle \phi\vert \chi\rangle$$

$$\langle \psi\vert \phi\rangle = \langle \phi\vert \psi\rangle^* =: \overline{\langle \phi\vert \psi\rangle }$$ (167)

Def.: A vector space with scalar product and norm $\Vert\psi\Vert=\sqrt{\langle \psi\vert \psi\rangle }$ is called unitary space.

Def.: A sequence $\{\psi_n\}$ in a unitary space is called Cauchy sequence, if with all real number $\varepsilon>0$ there is an integer $N(\varepsilon)$ such that for all $n,m>N(\varepsilon)$, $\langle \psi_n\vert \psi_m\rangle <\varepsilon)$ holds.

Def.: A unitary space X is called complete, if each Cauchy sequence in X converges to a vector $\psi \in X$.

Def.: A Hilbert space is a complete unitary space.