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In this chapter, we are interested in presenting a spectral decomposition of the Sobolev Space \(H^1(\mathbb {R}^n)\) . To this end, we provide a thorough construction stating the foundations of functional calculus for operators and present a proof of the Spectral Theorem for unbounded self-adjoint operators. Despite this last mentioned result is a classical result in functional analysis, there are few references that build all arguments from scratch, which we make here.

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Spectral Decomposition of \(H^1(\mathbb {R}^n)\)

  • 章节

  • 发表日期：

    2025年

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  • César Augusto Rubim,
  • Eduard Toon

摘要

In this chapter, we are interested in presenting a spectral decomposition of the Sobolev Space \(H^1(\mathbb {R}^n)\) . To this end, we provide a thorough construction stating the foundations of functional calculus for operators and present a proof of the Spectral Theorem for unbounded self-adjoint operators. Despite this last mentioned result is a classical result in functional analysis, there are few references that build all arguments from scratch, which we make here.