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Definition

A Hilbert space is a real or complex inner product space that is also a complete metric space with respect to the distance function induced by the inner product.

To say that a complex vector space H is a complex inner product space means that there is an inner product ⟨ x, y ⟩ {\displaystyle \langle x,y\rangle } associating a complex number to each pair of elements x, y {\displaystyle x,y} of H that satisfies the following properties:

It follows from properties 1 and 2 that a complex inner product is antilinear, also called conjugate linear, in its second argument, meaning that ⟨ x, a y 1 + b y 2 ⟩ = a ¯ ⟨ x, y 1 ⟩ + b ¯ ⟨ x, y 2 ⟩. {\displaystyle \langle x,ay_{1}+by_{2}\rangle ={\bar {a}}\langle x,y_{1}\rangle +{\bar {b}}\langle x,y_{2}\rangle \,.}

A real inner product space is defined in the same way, except that H is a real vector space and the inner product takes real values. Such an inner product will be a bilinear map and (H, H, ⟨ ⋅, ⋅ ⟩) {\displaystyle (H,H,\langle \cdot,\cdot \rangle)} will form a dual system.

The norm is the real-valued function ‖ x ‖ = ⟨ x, x ⟩, {\displaystyle \|x\|={\sqrt {\langle x,x\rangle }}\,,} and the distance d {\displaystyle d} between two points x, y {\displaystyle x,y} in H is defined in terms of the norm by d (x, y) = ‖ x − y ‖ = ⟨ x − y, x − y ⟩. {\displaystyle d(x,y)=\|x-y\|={\sqrt {\langle x-y,x-y\rangle }}\,.}

That this function is a distance function means firstly that it is symmetric in x {\displaystyle x} and y, {\displaystyle y,} secondly that the distance between x {\displaystyle x} and itself is zero, and otherwise the distance between x {\displaystyle x} and y {\displaystyle y} must be positive, and lastly that the triangle inequality holds, meaning that the length of one leg of a triangle xyz cannot exceed the sum of the lengths of the other two legs: d (x, z) ≤ d (x, y) + d (y, z). {\displaystyle d(x,z)\leq d(x,y)+d(y,z)\,.}

This last property is ultimately a consequence of the more fundamental Cauchy–Schwarz inequality, which asserts | ⟨ x, y ⟩ | ≤ ‖ x ‖ ‖ y ‖ {\displaystyle \left|\langle x,y\rangle \right|\leq \|x\|\|y\|} with equality if and only if x {\displaystyle x} and y {\displaystyle y} are linearly dependent.

With a distance function defined in this way, any inner product space is a metric space, and sometimes is known as a Hausdorff pre-Hilbert space. Any pre-Hilbert space that is additionally also a complete space is a Hilbert space.

The completeness of H is expressed using a form of the Cauchy criterion for sequences in H: a pre-Hilbert space H is complete if every Cauchy sequence converges with respect to this norm to an element in the space. Completeness can be characterized by the following equivalent condition: if a series of vectors ∑ k = 0 ∞ u k {\displaystyle \sum _{k=0}^{\infty }u_{k}} converges absolutely in the sense that ∑ k = 0 ∞ ‖ u k ‖ < ∞, {\displaystyle \sum _{k=0}^{\infty }\|u_{k}\|<\infty \,,} then the series converges in H, in the sense that the partial sums converge to an element of H.

As a complete normed space, Hilbert spaces are by definition also Banach spaces. As such they are topological vector spaces, in which topological notions like the openness and closedness of subsets are well defined. Of special importance is the notion of a closed linear subspace of a Hilbert space that, with the inner product induced by restriction, is also complete (being a closed set in a complete metric space) and therefore a Hilbert space in its own right.