바나흐 공간

노트

위키데이터

• ID : Q194397

말뭉치

1. His dichotomy theorem asserts that either every subspace of a given Banach space has many symmetries or the subspaces have only trivial symmetries.
2. A Banach space assumes that there is a norm on the space relative to which the space is complete, but it is not assumed that the norm is defined in terms of an inner product.
3. If, as a result of these inclusions, the Banach space coincides with its second dual, it is called reflexive.
4. A Banach space is said to be weakly complete if each weak Cauchy sequence in it weakly converges to an element of the space.
5. An incomplete normed space is not homeomorphic to any Banach space.
6. In infinite-dimensional Banach spaces unconditional convergence follows from absolute convergence but the converse is not true in any infinite-dimensional Banach space.
7. It is clearly written and provides accessible references to many techniques that are commonly used in contemporary research in Banach space theory.
8. if you need to know what a dentable Banach space is, you can find out here … .
9. E. A Banach space is called hereditarily indecomposable, if none of its subspaces is decomposable.
10. Let E be a Banach space.
11. Now we present a dichotomy concerning geometric structure of the unit ball in a Banach space, which covers this extremal case.
12. Assume that E is a Banach space with a basis and satisfies the assumptions of the Lemma.
13. Usually, the notion of Banach space is only used in the infinite dimensional setting, typically as a vector space of functions.
14. While a Hilbert space is always a Banach space, the converse need not hold.
15. Therefore, it is possible for a Banach space not to have a norm given by an inner product.
16. In a finite-dimensional Banach space X, any continuous functional defined on a closed bounded set M attains its infimum.
17. Maurice René Fréchet was the first to use the term "Banach space" and Banach in turn then coined the term "Fréchet space.
18. A Banach space is a complete normed space (X, || ⋅ ||).
19. By definition, the normed space (X, || ⋅ ||) is a Banach space if and only if (X, d) is a complete metric space, or said differently, if and only if the canonical metric d is a complete metric.
20. With this topology, every Banach space is a Baire space, although there are normed spaces that are Baire but not Banach.
21. To begin, recall that a Banach space is a complete normed linear space.
22. Essentially self-contained, this reference explores a fundamental aspect of Banach space theory.
23. A Banach space ℬ \mathcal{B} is both a vector space (over a normed field such as ℝ \mathbb{R} ) and a complete metric space, in a compatible way.
24. Then a Banach space is simply a vector space equipped with a complete norm.
25. Thus a Banach space may equivalently be defined as a vector space equipped with a complete homogeneous translation-invariant metric.
26. As such, ℒ p ( X ) \mathcal{L}^p(X) is a complete pseudonormed vector space; but we identify functions that are equal almost everywhere to make it into a Banach space.
27. The Eberlein-Smulian theorem is obtained in this section, as is the result due to Krein and Smulian that the closed convex hull of a weakly compact subset of a Banach space is itself weakly compact.
28. The goal of optional Section 2.9 is to obtain James's characterization of weakly compact subsets of a Banach space in terms of the behavior of bounded linear functionals.
29. The book revolves around two main broad questions: What is the best smoothness of a given Banach space, and its structural consequences?
30. We show that the space of real functions, which are integrable with respect to a measure with values in a Banach space , has (D) property.
31. A Banach space is said to have property if every bounded linear operator is weakly compact for every Banach space whose dual does not contain an isomorphic copy of .
32. We show that every Banach space whose dual has property has property.
33. Given a vector measure with values in a Banach space , denotes the space of (classes of) real functions that are integrable with respect to in the sense of Bartle et al.
34. Then, is an s-Banach space when equipped with the norm .

메타데이터

위키데이터

• ID : Q194397