This chapter develops Hilbert spaces from the norm induced by an inner product and completeness, then studies orthogonal expansions, linear functionals, bounded linear operators, spectral theory and unbounded operators.
Contents
- 13.1 Definitions and examples
- 13.2 Expansion theorems
- 13.3 Linear functionals
- 13.4 Bounded linear operators
- 13.5 Spectral theory
- 13.6 Unbounded operators
Concept index
Terms by section. Links lead to the brown underlined definitions in the Chinese reading text.
13.1 Definitions and examples
13.2 Expansion theorems
- subspace子空间
- closure闭包
- subspace generated by生成的子空间
- separable可分的
- complete orthonormal set完备正交标准集
- orthonormal basis正交标准基
- Bessel’s inequality贝塞尔不等式
- Hermite polynomials厄米多项式
13.3 Linear functionals
13.4 Bounded linear operators
- bounded有界的
- operator norm算符范数
- multiplication operator乘法算符
- invertible可逆的
- inverse逆
- adjoint伴随
- hermitian厄米的
- projection operator投影算符
- idempotent operator幂等算符
- unitary酉的
- isometric等距的
13.5 Spectral theory
- eigenvalue特征值
- eigenvector特征向量
- regular value正则值
- spectrum谱
- point spectrum点谱
- continuous spectrum连续谱
- absolutely continuous绝对连续的