1. 1 By Dr. Talat Nazir Functional Analysis MTH 327

2. 2 Vector Spaces Normed Spaces Lecture # 1 MTH 327

3. 3 Lecture’s outline Vector Spaces Some Examples of Vector Spaces Normed Spaces Some Examples of Normed Spaces

4. 4 Functional Analysis Functional analysis is an abstract branch of mathematics that originated from classical analysis. The impetus came from applications: problems related to ordinary and partial di ff erential equations, numerical analysis, calculus of variations, approximation theory, integral equations, and so on. In ordinary calculus, one dealt with limiting processes in finite-dimensional vector spaces (R or R n ), but problems arising in the above applications required a calculus in spaces of functions (which are infinite-dimensional vector spaces).

5. Vector Space

6. 6 Examples:

7. 7 More Examples:

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9. 9 Normed Spaces

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14. 14 More Examples of Normed Spaces:

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16. 16 Some Problems:

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