CT951
发表于 2025-3-21 16:41:37
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GRAVE
发表于 2025-3-21 23:22:25
Dimensionality,Let . be a vector space and let . be a subset of it. We say . is . if for every finite subset {.,…, .} of ., the equation.holds if and only if . = . = ⋯ = . = 0. A (finite) sum like the one in (2.1) is called a . of .,…, ..
evince
发表于 2025-3-22 04:08:53
New Banach Spaces from Old,Let . be a vector space and . a subspace of it. Say that two elements . and . of . are ., . ~ ., if . − . ∈ .. This is an equivalence relation on .. The coset of . under this relation is the set..Let . be the collection of all these cosets. If we set.then . is a vector space with these operations.
自作多情
发表于 2025-3-22 05:33:58
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dominant
发表于 2025-3-22 12:01:55
The Uniform Boundedness Principle,The . says that a complete metric space cannot be the union of a countable number of nowhere dense sets. This has several very useful consequences. One of them is the Uniform Boundedness Principle (U.B.P.) also called the ..
photopsia
发表于 2025-3-22 15:50:24
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Extort
发表于 2025-3-22 19:27:23
Dual Spaces,The idea of duality, and the associated notion of adjointness, are important in functional analysis. We will identify the spaces .* for some of the standard Banach spaces.
GLOSS
发表于 2025-3-23 00:06:08
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一夫一妻制
发表于 2025-3-23 04:05:25
The Second Dual and the Weak* Topology,The dual of .* is another Banach space .**. This is called the . or the . of .. Let . be the map from . into .** that associates with . ∈ . the element . ∈ .** defined as. Then . is a linear map and ‖.‖ = ‖.‖. (See (9.2).) Thus . is an . and we can regard . as a subspace of .**.
mercenary
发表于 2025-3-23 07:53:21
Orthonormal Bases,A subset . in a Hilbert space is said to be an . if 〈., .〉 = 0 for all ., . in . (. ∈ .), and ‖.‖ = 1 for all . in ..