B. Some Tools from Abstract Mathematical Analysis

Let V be a Hilbert space and
and
, the corresponding scalar product and norm, respectively.
A *linear form* (or *linear functional*) on is a function
such that,

(B.1) |

A linear form is *bounded* if there is a constant
such that,

(B.2) |

A *bilinear form* on is a function
, which is linear in each argument separately, i.e., such that, for all
and
,

(B.3) |

(B.4) |

The bilinear form
is said to be *symmetric* if,

(B.5) |

*bounded* if there is a constant
such that,

(B.6) |

and if there is a constant such that,

(B.7) |

The set of all bounded linear functionals on is called *dual space* of and denoted .
The norm in is given by,

(B.8) |

**Theorem I** (*Riesz's representation theorem*): Let be a Hilbert space with scalar product
.
For each bounded linear functional on there is an unique such that,

(B.9) |

Moreover,

(B.10) |

**Theorem II** (*Lax-Milgram lemma*): If the bilinear form
is bounded and -elliptic in the Hilbert space , and is bounded linear form in , than there exists a unique vector such that,

and,

(B.12) |

**Theorem III**: Assume that
is a symmetric, -elliptic bilinear form and that is a bounded linear form on the Hilbert space .
Than satisfies (B.11) if and only if,

where | (B.13) |

H. Ceric: Numerical Techniques in Modern TCAD