structural:expansion
Motivating example: Euclidean vector space

One of the most familiar examples of a Hilbert space is the Euclidean vector space consisting of three-dimensional vectors, denoted by R, and equipped with the dot product. The dot product takes two vectors x and y, and produces a real number x ⋅ y. If x and y are represented in Cartesian coordinates, then the dot product is defined by (x 1 x 2 x 3) ⋅ (y 1 y 2 y 3) = x 1 y 1 + x 2 y 2 + x 3 y 3. {\displaystyle {\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\end{pmatrix}}\cdot {\begin{pmatrix}y_{1}\\y_{2}\\y_{3}\end{pmatrix}}=x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}\,.}

The dot product satisfies the properties

An operation on pairs of vectors that, like the dot product, satisfies these three properties is known as a (real) inner product. A vector space equipped with such an inner product is known as a (real) inner product space. Every finite-dimensional inner product space is also a Hilbert space. The basic feature of the dot product that connects it with Euclidean geometry is that it is related to both the length (or norm) of a vector, denoted ‖ x ‖, and to the angle θ between two vectors x and y by means of the formula x ⋅ y = ‖ x ‖ ‖ y ‖ cos θ. {\displaystyle \mathbf {x} \cdot \mathbf {y} =\left\|\mathbf {x} \right\|\left\|\mathbf {y} \right\|\,\cos \theta \,.}

Multivariable calculus in Euclidean space relies on the ability to compute limits, and to have useful criteria for concluding that limits exist. A mathematical series ∑ n = 0 ∞ x n {\displaystyle \sum _{n=0}^{\infty }\mathbf {x} _{n}} consisting of vectors in R is absolutely convergent provided that the sum of the lengths converges as an ordinary series of real numbers: ∑ k = 0 ∞ ‖ x k ‖ < ∞. {\displaystyle \sum _{k=0}^{\infty }\|\mathbf {x} _{k}\|<\infty \,.}

Just as with a series of scalars, a series of vectors that converges absolutely also converges to some limit vector L in the Euclidean space, in the sense that ‖ L − ∑ k = 0 N x k ‖ → 0 as N → ∞. {\displaystyle {\Biggl \|}\mathbf {L} -\sum _{k=0}^{N}\mathbf {x} _{k}{\Biggr \|}\to 0\quad {\text{as }}N\to \infty \,.}

This property expresses the completeness of Euclidean space: that a series that converges absolutely also converges in the ordinary sense.

Hilbert spaces are often taken over the complex numbers. The complex plane denoted by C is equipped with a notion of magnitude, the complex modulus | z |, which is defined as the square root of the product of z with its complex conjugate: | z | 2 = z z ¯. {\displaystyle |z|^{2}=z{\overline {z}}\,.}

If z = x + iy is a decomposition of z into its real and imaginary parts, then the modulus is the usual Euclidean two-dimensional length: | z | = x 2 + y 2. {\displaystyle |z|={\sqrt {x^{2}+y^{2}}}\,.}

The inner product of a pair of complex numbers z and w is the product of z with the complex conjugate of w: ⟨ z, w ⟩ = z w ¯. {\displaystyle \langle z,w\rangle =z{\overline {w}}\,.}

This is complex-valued. The real part of ⟨ z, w ⟩ gives the usual two-dimensional Euclidean dot product.

A second example is the space C whose elements are pairs of complex numbers z = (z, z). Then the inner product of z with another such vector w = (w, w) is given by ⟨ z, w ⟩ = z 1 w 1 ¯ + z 2 w 2 ¯. {\displaystyle \langle z,w\rangle =z_{1}{\overline {w_{1}}}+z_{2}{\overline {w_{2}}}\,.}

The real part of ⟨ z, w ⟩ is then the two-dimensional Euclidean dot product. This inner product is Hermitian symmetric, which means that the result of interchanging z and w is the complex conjugate: ⟨ w, z ⟩ = ⟨ z, w ⟩ ¯. {\displaystyle \langle w,z\rangle ={\overline {\langle z,w\rangle }}\,.}