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tma4145:2019h:log [2019-07-12]
franzl
tma4145:2019h:log [2019-10-08] (nåværende versjon)
franzl
Linje 8: Linje 8:
   * 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.   * 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. Normal, unitary and self-adjoint operators. See Ch 5.3 in LN. 
 +
 +''​Eleventh week (week 44)''​
 +
 +  * Monday: Linear independence,​ Hamel bases, dimension, Schauder bases, (infinite) series in vector spaces, orthogonal sets and the closest point property, See Ch 6.1-6.3 in LN.
 +
 +  * Tuesday: The closest point property, Gram-Schmidt orthogonalization,​ convergence of series in Hilbert spaces, orthonormal bases, maximal orthonormal sequences, the Fourier series theorem. See Ch. 6.3-6.4 in LN.
 +
 +''​Twelfth week (week 45)''​
 +
 +  * Monday: Equivalent norms. Linear transformations on finite-dimensional vector spaces and their matrix representation. Null-space, column space and row space of a matrix. See Ch. 6.5 and 7.1 in LN.
 +
 +  * Tuesday: The rank-nullity theorem and its consequences. Eigenvalues and eigenvectors. Similarity transforms and Schur'​s lemma. The spectral theorem for Hermitian matrices. See Ch. 7.1-7.3 in LN.
 +
 +''​Thirteenth week (week 46)''​
 +
 +  * Monday: The spectral theorem. Positive and semi-positive definiteness. Singular value decomposition. See Ch. 7.4-7.5 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.
2019-07-12, Franz Luef