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  2. Inner product space - Wikipedia

    en.wikipedia.org/wiki/Inner_product_space

    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 } .

  3. Polarization identity - Wikipedia

    en.wikipedia.org/wiki/Polarization_identity

    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 ...

  4. Frobenius inner product - Wikipedia

    en.wikipedia.org/wiki/Frobenius_inner_product

    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

  5. Hilbert space - Wikipedia

    en.wikipedia.org/wiki/Hilbert_space

    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]

  6. Cosine similarity - Wikipedia

    en.wikipedia.org/wiki/Cosine_similarity

    In data analysis, cosine similarity is a measure of similarity between two non-zero vectors defined in an inner product space. Cosine similarity is the cosine of the angle between the vectors; that is, it is the dot product of the vectors divided by the product of their lengths. It follows that the cosine similarity does not depend on the ...

  7. Projection (linear algebra) - Wikipedia

    en.wikipedia.org/wiki/Projection_(linear_algebra)

    All these formulas also hold for complex inner product spaces, provided that the conjugate transpose is used instead of the transpose. Further details on sums of projectors can be found in Banerjee and Roy (2014). [9] Also see Banerjee (2004) [10] for application of sums of projectors in basic spherical trigonometry.

  8. Indefinite inner product space - Wikipedia

    en.wikipedia.org/wiki/Indefinite_inner_product_space

    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 ...

  9. Exterior algebra - Wikipedia

    en.wikipedia.org/wiki/Exterior_algebra

    For ⁠ ⁠ a finite-dimensional space, an inner product (or a pseudo-Euclidean inner product) on ⁠ ⁠ defines an isomorphism of with ⁠ ⁠, and so also an isomorphism of () with ⁠ ⁠. The pairing between these two spaces also takes the form of an inner product.