Titlebook: Introduction to Functional Analysis; Christian Clason Textbook 20201st edition Springer Nature Switzerland AG 2020 normed vector space.lin

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Quotient SpacesQuotient spaces are formed by collecting elements of a normed vector space into equivalence classes. This chapter covers results on these that are needed in later chapters.

歪曲道理

Linear Functionals and Dual SpacesContinuous linear functionals on a normed vector space generalize extracting components of finite-dimensional vectors and, collectively, form the dual space; these concepts yield crucial tools in functional analysis. This chapter gives the characterization of such functional on sequence, function, and quotient spaces.

起草

The Hahn–Banach TheoremThe Hahn–Banach theorem is another fundamental principle of functional analysis, which allows extending continuous linear functionals on a subspace while preserving continuity and linearity. An alternative version allows the separation of convex sets by hyperplanes. This chapter covers both versions together with their most important consequences.

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Compact OperatorsCompact operators map weakly convergent sequences to sequences converging in norm; in this way, they preserve further useful properties of finite-dimensional operators. Some of these are shown in this chapter, including Schauder’s theorem.

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Infantry

https://doi.org/10.1007/978-3-030-52784-6normed vector space; linear operators; dual space; weak convergence; spectrum

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