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MathidiaMathematics Academy
Functional methods
Lesson 23 of 75
Standard practice

Hilbert spaces

Objective. Apply inner-product geometry and orthogonal expansion.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

A Hilbert space is a Banach space whose norm arises from an inner product ⟨·,·⟩ via ‖v‖=√⟨v,v⟩, giving access to geometric notions like orthogonality and projections that general Banach spaces lack. The inner product satisfies linearity in one argument, conjugate symmetry, and positive-definiteness, and the Cauchy-Schwarz inequality |⟨u,v⟩|≤‖u‖‖v‖ follows directly. Any orthonormal basis {eₙ} allows expanding vectors as v=Σ⟨v,eₙ⟩eₙ with Parseval's identity ‖v‖²=Σ|⟨v,eₙ⟩|², generalising Fourier series. For example, L²[0,2π] is a Hilbert space with orthonormal basis {e^{inx}/√(2π)}, giving the classical Fourier expansion of square-integrable functions.

Worked example

In the Hilbert space ℝ³ with the standard inner product, find ⟨u,v⟩ for u=(1,2,3) and v=(4,−1,2).

v⟩ for u=(1
Illustrated card showing v⟩ for u=(1
  1. 1Use ⟨u,v⟩ = u₁v₁+u₂v₂+u₃v₃
  2. 2Compute 1×4 = 4
  3. 3Compute 2×(−1) = −2 and 3×2 = 6
  4. 4Sum: 4 + (−2) + 6 = 8

Answer: 8

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Common mistake. Students assume any Banach space automatically has an inner product structure, but the sup-norm space C[0,1] is a Banach space with no inner product inducing that norm, since the parallelogram law ‖u+v‖²+‖u−v‖²=2‖u‖²+2‖v‖² fails for the sup norm in general.

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