Abstract
A set \( \rlap{--} \rlap{--} \mathfrak{X} \) is called a complex normed space if
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(1)
\( \rlap{--} \rlap{--} \mathfrak{X} \) is a vector space over the field ℂ1 of complex numbers
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(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:
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(i)
$$ \left\| {ax} \right\| = \left| a \right|\left\| x \right\|(\forall x \in x,\forall a \in {\mathbb{C}^1}) $$
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(ii)
$$ \left\| {x + y} \right\| \leqslant \left\| x \right\| + \left\| y \right\|(\forall x,y \in x) $$
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(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 \( {\lim _{m,n \to \infty }}\left\| {{x_n} - {x_m}} \right\| = 0 \)
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© 1991 Springer Science+Business Media Dordrecht
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Gorbachuk, V.I., Gorbachuk, M.L. (1991). Some Information from the Theory of Linear Operators. In: Boundary Value Problems for Operator Differential Equations. Mathematics and Its Applications (Soviet Series), vol 48. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-3714-0_1
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DOI: https://doi.org/10.1007/978-94-011-3714-0_1
Publisher Name: Springer, Dordrecht
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