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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 } .
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 ...
The Gram-Schmidt theorem, together with the axiom of choice, guarantees that every vector space admits an orthonormal basis. This is possibly the most significant use of orthonormality, as this fact permits operators on inner-product spaces to be discussed in terms of their action on the space's orthonormal basis vectors. What results is a deep ...
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 ...
Given a finite-dimensional vector space over a field with a symmetric bilinear form (the inner product, [b] e.g., the Euclidean or Lorentzian metric) : , the geometric algebra of the quadratic space (,) is the Clifford algebra (,) , an element of which is called a multivector.
Using the group structure, any inner product on the tangent space at the identity (or any other particular tangent space) can be transported to all other tangent spaces to define a Riemannian metric. Formally, given an inner product g e on the tangent space at the identity, the inner product on the tangent space at an arbitrary point p is ...
In linear algebra, a frame of an inner product space is a generalization of a basis of a vector space to sets that may be linearly dependent. In the terminology of signal processing , a frame provides a redundant, stable way of representing a signal . [ 1 ]
The Gram matrix is symmetric in the case the inner product is real-valued; it is Hermitian in the general, complex case by definition of an inner product. The Gram matrix is positive semidefinite, and every positive semidefinite matrix is the Gramian matrix for some set of vectors. The fact that the Gramian matrix is positive-semidefinite can ...