#### 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.