Week

Topics to be Covered

1

1.1 Metric Space

1.2 Further Examples of Metric Spaces

1.3 Open Set, Closed Set, Neighborhood

1.4 Convergence, Cauchy Sequence, Completeness

2

2.1 Vector Space

2.2 Normed Space. Banach Space

2.3 Further Properties of Normed Spaces

2.4 Finite Dimensional Normed Spaces and Subspaces

3

2.5 Compactness and Finite Dimension

2.6 Linear Operators

2.7 Bounded and Continuous Linear Operators

2.8 Linear Functionals

4

2.9 Linear Operators and Functionals on Finite Dimensional Spaces

2.10 Normed Spaces of Operators. Dual Space

3.1 Inner Product Space. Hilbert Space

3.2 Further Properties of Inner Product Spaces

3.3 Orthogonal Complements and Direct Sums

5

3.4 Orthonormal Sets and Sequences

3.8 Representation of Functionals on Hilbert Spaces

3.9 Hilbert Adjoint Operator

3.10 Self-Adjoint, Unitary and Normal Operators

6

4.1 Zorn's Lemma

4.2 Hahn-Banach Theorem

4.3 Hahn-Banch Theorem for Complex Vector Spaces and Normed Spaces

4.5 Adjoint Operator

4.6 Reflexive Spaces

7

4.7 Category Theorem. Uniform Boundedness Theorem

4.8 Strong and Weak Convergence

4.9 Convergence of Sequences of Operators and Functionals

4.12 Open Mapping Theorem

4.13 Closed Linear Operators. Closed Graph Theorem.