#### Lectures log

`First week (week 34)`

- Monday: Naive set theory; basic definitions and facts about sets, de Morgan's laws. Functions; range and domain of a function, injective, surjective and bijective functions. See Chapter 1.1-1.2 in Lecture Notes (LN).

- Tuesday: Left and right inverses, inverses for functions and their characterization in terms of injectivity, surjectivity and bijectivity. Cardinality and countable sets. See Chapter 1.2-1.3 in LN.

`Second week (week 35)`

- Monday: Supremum and infimum of sets and functions. See Chapter 1.3-1.4 in LN.

- Tuesday: Real (and complex) vector spaces, basic examples of vector spaces, subspaces and linear transformations between vector spaces. See Chapter 2.1 in LN.

`Third week (week 36)`

- Monday: Normed spaces,metric spaces, p-norms, Young's inequality, Hoelder's inequality. See Chapter 2.2 in LN.

- Tuesday: Triangle inequality for ||.||_p norms of real n-tuples, complex n-tuples sequences and functions.

`Fourth week (week 37)`

- Monday: Inner product spaces, properties of the inner product, Cauchy-Schwarz inequality. See Chapter 2.2-2.3 in LN.

- Tuesday: Examples of inner product spaces, Jordan-von Neumann's characterization of norms induced by inner products, polarization identities, orthogonality, Pythagoras' theorem, convergence of sequences in normed spaces. See Chapters 2.3 and 3.1 in LN.

`Fifth week (week 38)`

- Monday: Convergent sequences in metric and normed spaces, bounded subsets and limit points, Cauchy sequences. See Chapter 3.1 in LN.

- Tuesday: Cauchy sequences and completeness, completeness of (R,|.|), examples of Banach spaces and completeness of the space of d-tuples with the sup-norm. See Ch. 3.1-3.2 in LN.

`Sixth week (week 39)`

- Monday: Further examples of Banach spaces, complete subspaces, uniform convergence of sequences of continuous functions. Isometries and isomorphic vector spaces.

- Tuesday: Isometrically isomorphic normed spaces, embeddings, dense subsets, separability, Stone-Weierstrass theorem. See Ch 3.3 in LN.

`Seventh week (week 40)`

- Monday: Banach's fixed point theorem and finding fixed points by iteration. Newton's method as a fixed point iteration. LN: Chapter 3.4-3.5, 4.1.

- Tuesday: Applications of Banach fixed point theorem to integral equations and differential equations. LN: Chapter 3.5. Linear operators, continuous operators,

`Eighth week (week 41)`

- Monday: Bounded operators. Equivalence of boundedness and continuity for linear operators, Extension theorem, the vector space of bounded linear operators between normed spaces, See Ch 4.2 in LN.

- Tuesday: Vector space of bounded linear operators, operator norm, completeness if co-domains are Banach spaces, proof that the kernel of a bounded linear operator is closed, the range need not be. See Ch. 4.2 in LN.

`Ninth week (week 42)`

- Monday: Dual space, best approximation theorem. See Ch. 4.3-5.1 in LN.

- Tuesday: Best approximation theorem, orthogonal complements, the projection theorem, consequences of the projection theorem. See Ch. 5.1 in LN.

`Tenth week (week 43)`

- Monday: Examples of Hilbert spaces, Riesz' representation theorem, proof and examples. See Ch 5.2 in LN.

- Tuesday: Adjoint operators, properties of adjoint operators and examples. See Ch 5.3 in LN

`Eleventh week (week 44)`

- Monday: Normal, unitary and self-adjoint operators. See Ch 5.3 in LN.Linear independence See Ch 6.1 in LN.

- Tuesday: Hamel bases, dimension, Schauder bases, (infinite) series in vector spaces, orthogonal sets and the closest point property,The closest point property, Gram-Schmidt orthogonalization, convergence of series in Hilbert spaces, orthonormal bases, maximal orthonormal sequences, the Fourier series theorem. See Ch. 6.1-6.4 in LN.

`Twelfth week (week 45)`

- Monday: Equivalent norms. Linear transformations on finite-dimensional vector spaces and their matrix representation. See Ch. 6.5 and 7.1 in LN.

- Tuesday: Null-space, column space and row space of a matrix. The rank-nullity theorem and its consequences. See Ch. 7.1-7.2 in LN.

`Thirteenth week (week 46)`

- Monday: Eigenvalues and eigenvectors. Similarity transforms and Schur's lemma. The spectral theorem for Hermitian matrices. The spectral theorem. Positive and semi-positive definiteness. Singular value decomposition. See Ch. 7.3-7.5 in LN.

- Tuesday: SVD example, brief discussion on applications of SVDs, pseudoinverse of a linear transformation, calculating the pseudoinverse of a matrix. See Ch. 7.5-7.6 in LN.