This appendix summarizes some concepts from functional analysis. The concepts are part of the mathematical background required for understanding this book. Mathematical peculiarities not relevant in this context are omitted. Instead, at the end of the appendix references to more detailed treatments are given.

A *linear space* (or *vector space*) over a field *F* is a set *R* with elements (*vectors*) **f, g, h, …** equipped with two operations:

*scalar multiplication*(*α***f**with*α*∈*F*):*F*×*R*→*R*

Usually, the field *F* is the set of real numbers, or the set of complex numbers. The addition and the multiplication operation must satisfy the following axioms:

- (a)
**f**+**g**=**g**+**f** - (b) (
**f**+**g**) +**h**=**f**+ (**g**+**h**) - (c) a so-called
*zero element***0**∈*R*exists such that**f**+**0**=**f** - (d) a
*negative element***−f**exists for each**f**such that**f**+ ( −**f**) =**0** - (e)
*α*(**f**+**g**) =*α***f**+*α***g** - (f) (
*α*+*β*)**f**=*α***f**+*β***f** - (g) (
*α**β*)**f**=*α*(*β*)**f** - (h) 1
**f**=**f**

A *linear subspace S* of a linear space *R* is a subset of *R* which itself is linear. A condition sufficient and necessary for a subset *S* ⊂ *R* to be linear is that *α***f** + *β***g** ∈ *S* for all **f, g** ∈ *S* and for all *α*, *β* ∈ *F*.

**Examples**

is the ...

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