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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 } .
The inner product of two vectors over the field of complex numbers is, in general, a complex number, and is sesquilinear instead of bilinear. An inner product space is a normed vector space, and the inner product of a vector with itself is real and positive-definite.
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]
In mathematics, the L p spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces.They are sometimes called Lebesgue spaces, named after Henri Lebesgue (Dunford & Schwartz 1958, III.3), although according to the Bourbaki group (Bourbaki 1987) they were first introduced by Frigyes Riesz ().
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).
Then there is a unique Hilbert space of functions on X for which K is a reproducing kernel. Proof. For all x in X, define K x = K(x, ⋅ ). Let H 0 be the linear span of {K x : x ∈ X}. Define an inner product on H 0 by
Every inner product space is also a normed space. A normed space underlies an inner product space if and only if it satisfies the parallelogram law, or equivalently, if its unit ball is an ellipsoid. Angles between vectors are defined in inner product spaces. A Hilbert space is defined as a complete inner product space. (Some authors insist ...
On a Krein space, the Hilbert inner product is positive definite, giving the structure of a Hilbert space (under a suitable topology). Under the weaker constraint K ± ⊂ K ± 0 {\displaystyle K_{\pm }\subset K_{\pm 0}} , some elements of the neutral subspace K 0 {\displaystyle K_{0}} may still be neutral in the Hilbert inner product, but many ...