Some Information from the Theory of Linear Operators

Abstract

A set $\rlap{--} \rlap{--} \mathfrak{X}$ is called a complex normed space if

• (1) $\rlap{--} \rlap{--} \mathfrak{X}$ is a vector space over the field ℂ¹ of complex numbers

• (2) for each element $x \in \rlap{--} X$ there is defined a non-negative number $\left\| x \right\|$ (the norm of x)possessing the following properties:

• (i)
  $$ \left\| {ax} \right\| = \left| a \right|\left\| x \right\|(\forall x \in x,\forall a \in {\mathbb{C}^1}) $$
• (ii)
  $$ \left\| {x + y} \right\| \leqslant \left\| x \right\| + \left\| y \right\|(\forall x,y \in x) $$

• (iii) $\left\| x \right\| = 0$ if and only if x=0.

A sequence ${x_n} \in x(n \in \mathbb{N} = \left\{ {1,2, \ldots } \right\})$ is said to converge in X to an element x if ${\lim _{n \to \infty }}\left\| {{x_n} - x} \right\| = 0$. A sequence ${x_n}(n \in \mathbb{N})$ from X is called fundamental if ${\lim _{m,n \to \infty }}\left\| {{x_n} - {x_m}} \right\| = 0$