Search results
Results From The WOW.Com Content Network
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 } .
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).
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 ...
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.
Let be Hermitian on inner product space with dimension , with spectrum ordered in ... Proof. By the min-max theorem, it suffices to show that any ...
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 (,), ...
If a normal operator T on a finite-dimensional real [clarification needed] or complex Hilbert space (inner product space) H stabilizes a subspace V, then it also stabilizes its orthogonal complement V ⊥. (This statement is trivial in the case where T is self-adjoint.) Proof. Let P V be the orthogonal projection onto V.
The quotient space of by the vector subspace is an inner product space with the inner product defined by +, + := (),,, which is well-defined due to the Cauchy–Schwarz inequality. The Cauchy completion of A / I {\displaystyle A/I} in the norm induced by this inner product is a Hilbert space, which we denote by H {\displaystyle H} .