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In mathematics, the Hilbert metric, also known as the Hilbert projective metric, is an explicitly defined distance function on a bounded convex subset of the n-dimensional Euclidean space R n. It was introduced by David Hilbert ( 1895 ) as a generalization of Cayley's formula for the distance in the Cayley–Klein model of hyperbolic geometry ...
With a distance function defined in this way, any inner product space is a metric space, and sometimes is known as a pre-Hilbert space. [6] Any pre-Hilbert space that is additionally also a complete space is a Hilbert space.
In mathematics, a metric space is a set together with a notion of distance between its elements, usually called points. The distance is measured by a function called a metric or distance function. [1] Metric spaces are the most general setting for studying many of the concepts of mathematical analysis and geometry.
The article on Hilbert spaces has several examples of inner product spaces, wherein the metric induced by the inner product yields a complete metric space. An example of an inner product space which induces an incomplete metric is the space C ( [ a , b ] ) {\displaystyle C([a,b])} of continuous complex valued functions f {\displaystyle f} and g ...
The Hilbert function, the Hilbert series and the Hilbert polynomial of a filtered algebra are those of the associated graded algebra. The Hilbert polynomial of a projective variety V in P n is defined as the Hilbert polynomial of the homogeneous coordinate ring of V.
The norm induced by this inner product is the Hilbert–Schmidt norm under which the space of Hilbert–Schmidt operators is complete (thus making it into a Hilbert space). [4] The space of all bounded linear operators of finite rank (i.e. that have a finite-dimensional range) is a dense subset of the space of Hilbert–Schmidt operators (with ...
Definition: Space is called a reproducing kernel Hilbert space if the evaluation functionals are continuous. Every RKHS has a special function associated to it, namely the reproducing kernel: Definition : Reproducing kernel is a function K : X × X → R {\displaystyle K:X\times X\to \mathbb {R} } such that
for all functions f ∈ L 2 (or, more typically, all f in a dense subset of L 2 such as a space of test functions, if the sequence {ψ k} is bounded). For given test functions, the relevant notion of convergence only corresponds to the topology used in . For example, in the Hilbert space L 2 (0,π), the sequence of functions