1
Setting and prerequisites
2
Axioms of an s-number sequence
3
The five canonical examples
▶
3.1
Approximation numbers \(a_n\)
3.2
Bernstein numbers \(b_n\)
3.3
Gelfand numbers \(c_n\)
3.4
Kolmogorov numbers \(d_n\)
3.5
Hilbert numbers \(h_n\)
4
The smallest and the largest s-numbers
5
Basic auxiliary results
6
Singular value decomposition
▶
6.1
Approximable operators
6.2
Schmidt representation
6.3
Compact \(\Leftrightarrow \) approximable on Hilbert spaces
6.4
All s-numbers equal the singular values in finite dimension
7
Inequalities between s-numbers
▶
7.1
John’s ellipsoid and the projection theorems
7.2
Approximation versus Gelfand and Kolmogorov numbers
7.3
The maximal difference theorem
8
Examples
▶
8.1
The identity \(\mathrm{id} : \ell _q^m \to \ell _p^m\)
8.2
Diagonal operators
8.3
The inclusion \(\ell _1 \to \ell _\infty \)
9
Injective and surjective s-numbers
10
Entropy numbers
11
Entropy numbers versus s-numbers
▶
11.1
Covering estimates
11.2
Compactness criteria
11.3
Entropy bound by Gelfand and Kolmogorov numbers
11.4
Entropy bound by Hilbert numbers
References
Dependency graph
Blueprint:
s-Numbers of Bounded Linear Operators
between Banach Spaces
Mario Ullrich
1
Setting and prerequisites
2
Axioms of an s-number sequence
3
The five canonical examples
3.1
Approximation numbers \(a_n\)
3.2
Bernstein numbers \(b_n\)
3.3
Gelfand numbers \(c_n\)
3.4
Kolmogorov numbers \(d_n\)
3.5
Hilbert numbers \(h_n\)
4
The smallest and the largest s-numbers
5
Basic auxiliary results
6
Singular value decomposition
6.1
Approximable operators
6.2
Schmidt representation
6.3
Compact \(\Leftrightarrow \) approximable on Hilbert spaces
6.4
All s-numbers equal the singular values in finite dimension
7
Inequalities between s-numbers
7.1
John’s ellipsoid and the projection theorems
7.2
Approximation versus Gelfand and Kolmogorov numbers
7.3
The maximal difference theorem
8
Examples
8.1
The identity \(\mathrm{id} : \ell _q^m \to \ell _p^m\)
8.2
Diagonal operators
8.3
The inclusion \(\ell _1 \to \ell _\infty \)
9
Injective and surjective s-numbers
10
Entropy numbers
11
Entropy numbers versus s-numbers
11.1
Covering estimates
11.2
Compactness criteria
11.3
Entropy bound by Gelfand and Kolmogorov numbers
11.4
Entropy bound by Hilbert numbers
References