Hilbert Spaces

Peter A. Loeb
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Peter A. Loeb: University of Illinois

Chapter Chapter 8 in Real Analysis, 2016, pp 127-145 from Springer

Abstract: Abstract A Hilbert space is, among other things, a linear space with either real or complex scalars; it is stable with respect to addition and scalar multiplication. To avoid repetition, we will work with the complex case, which will include the real case. That is, for a complex number $$z = x + iy$$ (x and y real), the conjugate is $$\overline{z} = x - iy$$ . For a real number a, the conjugate $$\overline{a} = a$$ . Therefore, if the scalar field is just the real numbers, then the conjugation operation $$a\mapsto \overline{a}$$ is just the identity operation on $$\mathbb{R}$$ . For this reason, we can treat both the real and complex scalar cases at the same time. Here is the definition of the space.

Keywords: Hilbert Space; Complex Scalars; Identity Operation; Scalar Field; Unordered Sum (search for similar items in EconPapers)
Date: 2016
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DOI: 10.1007/978-3-319-30744-2_8

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