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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 ( H , H , ⋅ , ⋅ ) {\displaystyle (H,H,\langle \cdot ,\cdot \rangle )} will form a dual system .
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 } .
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 ...
This example used the standard inner product, which is the map := ¯, but if a different inner product is used, such as := ¯ where is any Hermitian positive-definite matrix, or if a different orthonormal basis is used then the transformation matrices, and thus also the above formulas, will be different.
Inner product space – Generalization of the dot product; used to define Hilbert spaces; Minkowski distance – Mathematical metric in normed vector space; Normed vector space – Vector space on which a distance is defined; Polarization identity – Formula relating the norm and the inner product in a inner product space
This definition applies to a Banach space, but of course other types of space exist as well; for example, topological vector spaces include Banach spaces, but can be more general. [12] [13] On the other hand, Banach spaces include Hilbert spaces, and it is these spaces that find the greatest application and the richest theoretical results. [14]
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 following, seemingly weaker, definition is also equivalent: Definition 3. A unitary operator is a bounded linear operator U : H → H on a Hilbert space H for which the following hold: the range of U is dense in H, and; U preserves the inner product of the Hilbert space, H. In other words, for all vectors x and y in H we have: