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# 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,

 (B.11)

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)

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H. Ceric: Numerical Techniques in Modern TCAD