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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 , .
The requirement that is a positive-definite inner product then says exactly that this matrix-valued function is a symmetric positive-definite matrix at . In terms of the tensor algebra , the Riemannian metric can be written in terms of the dual basis { d x 1 , … , d x n } {\displaystyle \{dx^{1},\ldots ,dx^{n}\}} of the cotangent bundle as
The integral is absolutely convergent and the Petersson inner product is a positive definite Hermitian form. For the Hecke operators T n {\displaystyle T_{n}} , and for forms f , g {\displaystyle f,g} of level Γ 0 {\displaystyle \Gamma _{0}} , we have:
Besides these basic concepts, linear algebra also studies vector spaces with additional structure, such as an inner product. The inner product is an example of a bilinear form, and it gives the vector space a geometric structure by allowing for the definition of length and angles. Formally, an inner product is a map
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, 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.
The U.S. Environmental Protection Agency on Wednesday granted California its request to enforce vehicle emissions standards stricter than federal rules, including the state's ban on sales of new ...
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.