| Begge sider forrige revisjon
Forrige revisjon
Neste revisjon
|
Forrige revisjon
Neste revisjon
Begge sider neste revisjon
|
tma4145:2019h:log [2019-08-26]
franzl |
tma4145:2019h:log [2019-10-08]
franzl |
| ''Second week (week 35)'' | ''Second week (week 35)'' |
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| * Monday: Countable and uncountable sets, Cantor's diagonal argument. Supremum and infimum of sets and functions. See Chapter 1.3-1.4 in LN. | * Monday: Supremum and infimum of sets and functions. See Chapter 1.3-1.4 in LN. |
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| * Tuesday: Supremum and infimum of functions and sequences, real (and complex) vector spaces, basic examples of vector spaces, subspaces and linear transformations between vector spaces. See Chapter 1.4-2.1 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. |
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| ''Third week (week 36)'' | ''Third week (week 36)'' |
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| * Monday: Normed spaces, metric spaces, open balls in metric spaces, interior and boundary points of a subset, open and closed subsets, closures of sets, p-norms, Young's inequality. See Chapter 2.2 in LN. | * Monday: Normed spaces,metric spaces, p-norms, Young's inequality, Hoelder's inequality. See Chapter 2.2 in LN. |
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| * Tuesday: Hölder's inequality, triangle inequality for ||.||_p norms of real n-tuples, complex n-tuples and sequences. | * Tuesday: Triangle inequality for ||.||_p norms of real n-tuples, complex n-tuples sequences and functions. |
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| ''Fourth week (week 37)'' | ''Fourth week (week 37)'' |
| * Monday: ||.||_p norms on function spaces, inner product spaces, properties of the inner product, Cauchy-Schwarz inequality. See Chapter 2.2-2.3 in LN. | * Monday: Inner product spaces, properties of the inner product, Cauchy-Schwarz inequality. See Chapter 2.2-2.3 in LN. |
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| * 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. | * 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. |
| * Monday: Convergent sequences in metric and normed spaces, bounded subsets and limit points, Cauchy sequences. See Chapter 3.1 in LN. | * Monday: Convergent sequences in metric and normed spaces, bounded subsets and limit points, Cauchy sequences. See Chapter 3.1 in LN. |
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| * Tuesday: Cauchy sequences and completeness, completeness of (R,|.|), examples of Banach spaces. See Ch. 3.1-3.2 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. |
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| ''Sixth week (week 39)'' | ''Sixth week (week 39)'' |
| * Monday: Further examples of Banach spaces, complete subspaces, uniform convergence of sequences of continuous functions. Isometries and isomorphic vector spaces. | * Monday: Further examples of Banach spaces, complete subspaces, uniform convergence of sequences of continuous functions. Isometries and isomorphic vector spaces. |
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| *Tuesday: Isometrically isomorphic normed spaces, embeddings, dense subsets, separability, Stone-Weierstrass theorem, completions. See Ch 3.3 in LN. | *Tuesday: Isometrically isomorphic normed spaces, embeddings, dense subsets, separability, Stone-Weierstrass theorem. See Ch 3.3 in LN. |
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| ''Seventh week (week 40)'' | ''Seventh week (week 40)'' |
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| * Monday: Banach's fixed point theorem and finding fixed points by iteration. Newton's method as a fixed point iteration. LN: Chapter 3.4 and up to equation 3.13 in chapter 3.5. | * 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. |
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| * Tuesday: Applications of Banach fixed point theorem to integral equations and differential equations. LN: Chapter 3.5. | * Tuesday: Applications of Banach fixed point theorem to integral equations and differential equations. LN: Chapter 3.5. Linear operators, continuous operators, |
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| ''Eighth week (week 41)'' | ''Eighth week (week 41)'' |
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| * Monday: Linear operators, continuous operators, bounded operators. See Ch 4.1-4.2 in LN. | * 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. |
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| * Tuesday: Equivalence of boundedness and continuity for linear operators, proof that the kernel of a bounded linear operator is closed, the range need not be. 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. |
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| ''Ninth week (week 42)'' | ''Ninth week (week 42)'' |
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| * Monday: Extension theorem, the vector space of bounded linear operators between normed spaces, dual space, best approximation theorem. See Ch. 4.3-5.1 in LN. | * Monday: Dual space, best approximation theorem. See Ch. 4.3-5.1 in LN. |
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| * Tuesday: Best approximation theorem, orthogonal complements, the projection theorem, consequences of the projection theorem. See Ch. 5.1 in LN. | * Tuesday: Best approximation theorem, orthogonal complements, the projection theorem, consequences of the projection theorem. See Ch. 5.1 in LN. |