Functional Analysis 1

Lecture notes: FA_Skript.pdf (881 KB, english), FA_Skript_ger.pdf (387 KB, german)
Please note that the Closed Range Theorem (Theorem 5.8) was part of the exercises and hence it is not relevant for the exams (unless it was announced that way or included in the lecture).

Construction of the L^p

spaces and some properties: Lp-spaces.pdf (290 KB)
The spaces L^p(I)

, where

is a real interval, are defined, starting with $\mathcal L^p(I)$ . Then we show their Banach space property.
First, the case $1<p<\infty$ is considered, then also $p in {1,\infty}$ .
Of course, the definitions and proofs use methods of functional analysis.

Theory of closed operators: ClosedOperators.pdf (162 KB)
Taken out of an exercise. Contains the definition of a closed operator, two theorems and an example.

Theory of Fredholm operators: FredholmOperators.pdf (219 KB)
Taken out of an exercise. Contains the definition of a Fredholm operator, a lemma and the Atkinson's Theorem.

Some results of spectral theory: SpectralTheory.pdf (161 KB)
Summarizes the properties of spectral values etc. known from the lecture; without proofs.

Additional material on Dunford and Pettis integration: DunfordPettisInt.pdf (193 KB)

Most important theorems: FA1Theorems.pdf (209 KB)
The numbering of the unnamed theorems follows the lecture notes from above.
Note: This list only contains the theorems I deem most important. In particular, it does not contain everything relevant for the exams.

Contact

Every mistake found in the materials as well as any other comments can be sent to che.netzer(at)yahoo.de