is a type of mathematical space
is a real or complex inner product space (𝑉,𝐹,⟨·,·⟩) that is also a complete metric space with respect to the distance function (𝑑⟨·,·⟩) induced by the inner product (⟨·,·⟩)
if given an inner product (⟨·,·⟩), then the induced norm (||·||) is defined as:
- $∣∣ \cdot ∣ ∣_{⟨ \cdot,\cdot ⟩} = ⟨ \cdot,\cdot ⟩$
the tuple (𝑉,𝐹,⟨·,·⟩) is a Hilbert space, if (𝑉,𝐹,||·||_⟨·,·⟩) is a Banach space

Hilbert Spaces - Intro

Hilbert Spaces - Examples

	𝑉	𝐹	⟨·,·⟩	Description
Example #1	ℝ²	ℝ	$⟨ x, y ⟩ = \sum_{i = 1}^{2} x_{i} y_{i}$
Example #2	ℂ²	ℂ	$⟨ x, y ⟩ = \sum_{i = 1}^{2} \overline{x_{i}} y_{i}$	𝑥̅_𝑖 is the complex conjugate
Example #3	𝐿²(ℕ)	ℝ	$⟨ x, y ⟩ = \sum_{i = 1}^{\infty} x_{i} y_{i}$	𝐿²(ℕ) is Lp space with p = 2

	𝑉	𝐹	⟨·,·⟩	Description
Example #1	𝐶([0,1])	ℂ	$⟨ f, g ⟩ = \int_{0}^{1} \overline{f (t)} g (t) d t$	𝐶([0,1]) is the vector space of continuous functions defined on the unit interval [0,1] IS NOT A HILBERT SPACE BECAUSE IT FAILS COMPLETENESS