Advertisement

Measurable Functions on Hilbert Spaces

Chapter
  • 1.3k Downloads
Part of the Classics in Mathematics book series

Abstract

Consider a measurable Hilbert space (ℋ,𝓑) on which a measure μ is defined. Every continuous linear functional l(x) defined on ℋ is clearly 𝓑-measurable. It is known that if a sequence of continuous linear functionals l n (x) converges to a certain limit l(x) for all x then this limit will also be a continuous linear functional on ℋ. The situation, however becomes different if we require that l n (x) possess the limit not for all x but only on a set D such that μ(D)= 1. It is natural to refer to such limiting functions l(x) as 𝓑-measurable functionals. These functions, being limits of sequences of measurable functions will also be 𝓑-measurable. It follows from the relations
$$ \mathop{{\lim }}\limits_{{n \to \infty }} {l_n}\left( {\alpha x + \beta y} \right) = \alpha \mathop{{\lim }}\limits_{{n \to \infty }} {l_n}(x) + \beta \mathop{{\lim }}\limits_{{n \to \infty }} {l_n}(y) $$
that the domain of definition D l of functional l(x) is a linear manifold and that l(x) is a linear (additive and homogeneous) functional. (We assume that the functional l(x) is defined wherever the corresponding limit exists.) Hereafter we shall consider non-degenerate measures μ such that μ(L) = 0 for any proper subspace L of the space ℋ.

Keywords

Hilbert Space Measurable Function Orthogonal Polynomial Homogeneous Polynomial Linear Manifold 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Institute of Applied Mathematics and MechanicsAcademy of Sciences of the Ukrainian SSRDonetskUSSR
  2. 2.Institute of MathematicsAcademy of Sciences of the Ukrainian SSRKievUSSR

Personalised recommendations