Orthogonality

In this chapter, we will learn about:

  1. Inner products
  2. Complex inner products
  3. Inner product spaces
  4. Orthogonality
  5. Completeness
  6. Schwarz's inequality
  7. Complete orthonormal sets
  8. Projection theorem
  9. Linear functionals
  10. Parentheses versus brackets
  11. Natural isomorphisms
  12. Self-adjoint orthogonality
  13. Polarization
  14. Positive orthogonality
  15. Isometries
  16. Change of orthonormal basis
  17. Perpendicular projections
  18. Combinations of perpendicular projections
  19. Complexification
  20. Characterization of spectra
  21. Spectral theorem
  22. Normal orthogonality
  23. Orthogonal orthogonality
  24. Functions of orthogonality
  25. Polar decomposition
  26. Commutativity
  27. Self-adjoint orthogonality of rank one