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Questions and Answers
In functional analysis, which concept deals with the study of complete normed vector spaces?
In functional analysis, which concept deals with the study of complete normed vector spaces?
- Hilbert spaces
- Banach spaces (correct)
- Metric spaces
- Topological spaces
In functional analysis, what does the term 'Fredholm operator' refer to?
In functional analysis, what does the term 'Fredholm operator' refer to?
- Operator with closed range and closed kernel
- Unbounded operator with compact resolvent
- Self-adjoint operator on a Hilbert space
- Compact operator with finite-dimensional kernel and finite-codimensional range (correct)
True or false: Any compact operator in a Banach space has a nontrivial closed invariant subspace?
True or false: Any compact operator in a Banach space has a nontrivial closed invariant subspace?
True (A)
True or false: In functional analysis, the Arzelà –Ascoli theorem is used to characterize compact sets of functions?
True or false: In functional analysis, the Arzelà –Ascoli theorem is used to characterize compact sets of functions?
True or false: The Hahn-Banach theorem states that any bounded linear functional defined on a subspace of a normed vector space can be extended to the whole space?
True or false: The Hahn-Banach theorem states that any bounded linear functional defined on a subspace of a normed vector space can be extended to the whole space?
True or false: The Banach-Alaoglu theorem states that the closed unit ball in the dual space of a normed vector space is weak*-compact?
True or false: The Banach-Alaoglu theorem states that the closed unit ball in the dual space of a normed vector space is weak*-compact?
True or false: The Riesz representation theorem in functional analysis states that every bounded linear functional on a Hilbert space is represented by an inner product with a unique element in the space?
True or false: The Riesz representation theorem in functional analysis states that every bounded linear functional on a Hilbert space is represented by an inner product with a unique element in the space?
True or false: The open mapping theorem in functional analysis states that a continuous surjective linear mapping between Banach spaces is an open mapping?
True or false: The open mapping theorem in functional analysis states that a continuous surjective linear mapping between Banach spaces is an open mapping?
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