Forskjeller
Her vises forskjeller mellom den valgte versjonen og den nåværende versjonen av dokumentet.
Neste revisjon | Forrige revisjon Neste revisjon Begge sider neste revisjon | ||
tma4145:2019h:log [2019-07-12] franzl opprettet |
tma4145:2019h:log [2019-10-08] franzl |
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Linje 4: | Linje 4: | ||
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- | * Monday: Naive set theory; basic definitions and facts about sets, de Morgan' | + | * Monday: Naive set theory; basic definitions and facts about sets, de Morgan' |
* Tuesday: Left and right inverses, inverses for functions and their characterization in terms of injectivity, | * Tuesday: Left and right inverses, inverses for functions and their characterization in terms of injectivity, | ||
+ | '' | ||
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+ | * Monday: | ||
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+ | * Tuesday: | ||
+ | |||
+ | '' | ||
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+ | * Monday: | ||
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+ | * Tuesday: | ||
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+ | '' | ||
+ | * 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' | ||
+ | |||
+ | '' | ||
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+ | * 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, | ||
+ | |||
+ | '' | ||
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+ | * 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, | ||
+ | |||
+ | '' | ||
+ | |||
+ | * Monday: Banach' | ||
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+ | * Tuesday: Applications of Banach fixed point theorem to integral equations and differential equations. LN: Chapter 3.5. Linear operators, continuous operators, | ||
+ | |||
+ | '' | ||
+ | |||
+ | * Monday: | ||
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+ | * 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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+ | '' | ||
+ | |||
+ | * Monday: | ||
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+ | * Tuesday: Best approximation theorem, orthogonal complements, | ||
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+ | '' | ||
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+ | * Monday: Examples of Hilbert spaces, Riesz' representation theorem, proof and examples. See Ch 5.2 in LN. | ||
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+ | * Tuesday: Adjoint operators, properties of adjoint operators and examples. Normal, unitary and self-adjoint operators. See Ch 5.3 in LN. | ||
+ | |||
+ | '' | ||
+ | |||
+ | * Monday: Linear independence, | ||
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+ | * Tuesday: The closest point property, Gram-Schmidt orthogonalization, | ||
+ | |||
+ | '' | ||
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+ | * 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' | ||
+ | |||
+ | '' | ||
+ | |||
+ | * 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, |