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In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space [1] [2]) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar , often denoted with angle brackets such as in a , b {\displaystyle \langle a,b\rangle } .
In linear algebra, a branch of mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space. If a norm arises from an inner product then the polarization identity can be used to express this inner product entirely in terms of the norm. The ...
where , is the inner product.Examples of inner products include the real and complex dot product; see the examples in inner product.Every inner product gives rise to a Euclidean norm, called the canonical or induced norm, where the norm of a vector is denoted and defined by ‖ ‖:= , , where , is always a non-negative real number (even if the inner product is complex-valued).
Ptolemy's inequality holds more generally in any inner product space, [1] [9] and whenever it is true for a real normed vector space, that space must be an inner product space. [9] [10] For other types of metric space, the inequality may or may not be valid. A space in which it holds is called Ptolemaic.
This inner product can expressed in terms of the norm by using ... [proof 1] although it is also a consequence of ... space is defined as (,), ...
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 (,, , ) will form a dual system. [5]
Hadamard product (matrices) Hilbert–Schmidt inner product; Kronecker product; Matrix analysis; Matrix multiplication; Matrix norm; Tensor product of Hilbert spaces – the Frobenius inner product is the special case where the vector spaces are finite-dimensional real or complex vector spaces with the usual Euclidean inner product
Cauchy–Schwarz inequality – Mathematical inequality relating inner products and norms Hölder's inequality – Inequality between integrals in Lp spaces Mahler's inequality – inequality relating geometric mean of two finite sequences of positive numbers to the sum of each separate geometric mean Pages displaying wikidata descriptions as a ...