Book contents
- Frontmatter
- Contents
- Foreword
- Introduction
- 1 Inner product spaces
- 2 Normed spaces
- 3 Hilbert and Banach spaces
- 4 Orthogonal expansions
- 5 Classical Fourier series
- 6 Dual spaces
- 7 Linear operators
- 8 Compact operators
- 9 Sturm–Liouville systems
- 10 Green's functions
- 11 Eigenfunction expansions
- 12 Positive operators and contractions
- 13 Hardy spaces
- 14 Interlude: complex analysis and operators in engineering
- 15 Approximation by analytic functions
- 16 Approximation by meromorphic functions
- Appendix: square roots of positive operators
- References
- Answers to selected problems
- Afterword
- Index of notation
- Subject index
- Frontmatter
- Contents
- Foreword
- Introduction
- 1 Inner product spaces
- 2 Normed spaces
- 3 Hilbert and Banach spaces
- 4 Orthogonal expansions
- 5 Classical Fourier series
- 6 Dual spaces
- 7 Linear operators
- 8 Compact operators
- 9 Sturm–Liouville systems
- 10 Green's functions
- 11 Eigenfunction expansions
- 12 Positive operators and contractions
- 13 Hardy spaces
- 14 Interlude: complex analysis and operators in engineering
- 15 Approximation by analytic functions
- 16 Approximation by meromorphic functions
- Appendix: square roots of positive operators
- References
- Answers to selected problems
- Afterword
- Index of notation
- Subject index
Summary
The inner product spaces ℂn and ℓ2 share a further convenient property: informally speaking, any sequence in either of these spaces which looks convergent is convergent. To see what this means, recall the inner product space ℓ0 of Example 2.7. We saw there a sequence (xk) converging in ℓ2 but not in ℓ0. If we tried to carry out all our analysis in ℓ0 this phenomenon would definitely complicate matters: it would be like trying to do real analysis in ℚ instead of ℝ. Let us formulate the requirement of the existence of limits.
Definition Let (M,d) be a metric space. A sequence (xk) in M is a Cauchy sequence if, for every ε > 0, there exists an integer k0 such that k,l≥k0 implies that d(xk,xl) <ε.
(M,d) is a complete metric space if every Cauchy sequence in M converges to a limit in M.
Thus ℝ is a complete metric space with respect to its natural metric. So also is ℝ, for if (zk) is a Cauchy sequence in ℂ, then (Re zk) and (Im zk) are Cauchy sequences in ℝ. They thus have limits x,y∈ℝ, and we have zk → x + iy in ℂ.
Theorem ℂn (for n∈ℕ) and ℓ2 are complete metric spaces.
We recall our convention that metric terminology refers to the metric determined by the norm according to Theorem 2.3 for an inner product space.
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- An Introduction to Hilbert Space , pp. 21 - 30Publisher: Cambridge University PressPrint publication year: 1988