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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 , .
In mathematics, the interior product (also known as interior derivative, interior multiplication, inner multiplication, inner derivative, insertion operator, or inner derivation) is a degree −1 (anti)derivation on the exterior algebra of differential forms on a smooth manifold.
In mathematics, the Frobenius inner product is a binary operation that takes two matrices and returns a scalar.It is often denoted , .The operation is a component-wise inner product of two matrices as though they are vectors, and satisfies the axioms for an inner product.
In mathematics, the dot product or scalar product [note 1] is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors), and returns a single number. In Euclidean geometry , the dot product of the Cartesian coordinates of two vectors is widely used.
As with the direct and semidirect products, there is an external version of the Zappa–Szép product for groups which are not known a priori to be subgroups of a given group. To motivate this, let G = HK be an internal Zappa–Szép product of subgroups H and K of the group G.
Then C × is the internal direct product of the circle group T of unit complex numbers and the group R + of positive real numbers under multiplication. If n is odd, then the general linear group GL(n, R) is the internal direct product of the special linear group SL(n, R) and the subgroup consisting of all scalar matrices.
where is a bifunctor, the internal product functor defining a monoidal category. The isomorphism is natural in both X and Z. In other words, in a closed monoidal category, the internal Hom functor is an adjoint functor to the internal product functor. The object is called the internal Hom.
This kind of construction is called an inner semidirect product (also known as internal semidirect product [2]). Let us now consider the outer semidirect product. Given any two groups N and H and a group homomorphism φ : H → Aut( N ) , we can construct a new group N ⋊ φ H , called the outer semidirect product of N and H with respect to φ ...