Analysis Now pp 79-125 | Cite as

Hilbert Spaces

  • Gert K. Pedersen
Part of the Graduate Texts in Mathematics book series (GTM, volume 118)


The geometry of infinite-dimensional Banach spaces offers quite a few surprises from the viewpoint of finite-dimensional euclidean spaces. Thus, the unit ball may have corners, and closed convex sets may fail to have elements of minimal norm. Even more alienating, there may be no notion of perpendicular vectors and no good notion of a basis. By contrast, the Hilbert spaces are perfect generalizations of euclidean spaces, to the point of being almost trivial as geometrical objects. The deeper theory (and the fruitful applications) is, however, concerned with the operators on Hilbert space. Accordingly, we devote only a single section to Hilbert spaces as such, centered around the notions of sesquilinear forms, orthogonality, and self-duality. We then develop the elementary theory of bounded linear operator on a Hilbert space ℌ, i.e. we initiate the study of the Banach *-algebra B(ℌ)—to be continued in later chapters.


Hilbert Space Orthonormal Basis Compact Operator Toeplitz Operator Closed Subspace 
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Copyright information

© Springer Science+Business Media New York 1989

Authors and Affiliations

  • Gert K. Pedersen
    • 1
  1. 1.Mathematics InstituteUniversity of CopenhagenCopenhagen ØDenmark

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