The bra and ket vectors that we now use form a more general space than a Hilbert space'

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• The bra and ket vectors that we now use form a
  more general space than a Hilbert space.
• all realizable states correspond to ket vectors
  that can be normalized and thus form a Hilbert
  space.
• from The Principles of Quantum Mechanics by Paul
  Dirac (Physicist)

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Kinds of Convergence

• Convergence in the Mean fn? ?f ? if for any
  real e gt 0 there is an N(e) such that
• for all n gt N.
• Pointwise convergence For any x and any real e gt
  0 there is an N(e,x) such that
• 
  for all n gt N.
• Uniform convergenceFor any x and any real e gt 0
  there is an N(e) independent of x, such that
• 
  for all n gt N.

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Kinds of convergence

• Uniform convergence implies both pointwise and
  mean convergence
• Pointwise convergence does not imply either of
  the other two
• Q10.1 example of pointwise but not mean
  convergence.
• Fourier sequence for square wave for any x0 ? 0,
  we can include enough Fourier components to push
  the Gibbs ringing closer to zero than x0, but
  the max is always 1.18 convergence is pointwise
  everywhere except x0, but not uniform.
• Mean convergence does not imply either of the
  other two
• Fourier sequence for square wave converges only
  in the mean, always wrong at x0.