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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 , .
It is often called the inner product (or rarely the projection product) of Euclidean space, even though it is not the only inner product that can be defined on Euclidean space (see Inner product space for more). Algebraically, the dot product is the sum of the products of the corresponding entries of the two sequences of numbers.
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 .
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:
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 inner product of a Euclidean space is often called dot product and denoted x ⋅ y. This is specially the case when a Cartesian coordinate system has been chosen, as, in this case, the inner product of two vectors is the dot product of their coordinate vectors. For this reason, and for historical reasons, the dot notation is more commonly ...
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.
For example, a bilinear form is the same thing as a (0, 2)-tensor; an inner product is an example of a (0, 2)-tensor, but not all (0, 2)-tensors are inner products. In the (0, M ) -entry of the table, M denotes the dimensionality of the underlying vector space or manifold because for each dimension of the space, a separate index is needed to ...