an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product...

Hilbert space is a real or complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes...

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). It should...

indefinite inner product space ( K , ⟨ ⋅ , ⋅ ⟩ , J ) {\displaystyle (K,\langle \cdot ,\,\cdot \rangle ,J)} is an infinite-dimensional complex vector space K {\displaystyle...

specifically a Hilbert space, because the space is complete under the metric induced by the inner product. This inner product space is conventionally denoted...

2\|x\|^{2}+2\|y\|^{2}\quad {\text{ for all }}x,y.} In an inner product space, the norm is determined using the inner product: ‖ x ‖ 2 = ⟨ x , x ⟩ . {\displaystyle \|x\|^{2}=\langle...

vector space equipped with the usual dot product ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } (thus making it an inner product space), and let...

complete for this norm. An inner product space is a normed vector space whose norm is the square root of the inner product of a vector and itself. The...

conditions and called an inner product. Every inner product space is also a normed space. A normed space underlies an inner product space if and only if it satisfies...

Mirror (2022) Priam InnerSpace, a hard disk drive series by Priam Corporation in the 1980s Inner product space, a kind of vector space in linear algebra...

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...

generally in any inner product space, and whenever it is true for a real normed vector space, that space must be an inner product space. For other types...

bound on the absolute value of the inner product between two vectors in an inner product space in terms of the product of the vector norms. It is considered...

mathematics, particularly linear algebra, an orthonormal basis for an inner product space V {\displaystyle V} with finite dimension is a basis for V {\displaystyle...

defined over a set of labels S {\displaystyle S} in an inner product space with an inner product ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle...

of an inner product. Physical space can be modelled as a vector space which additionally has the structure of an inner product. The inner product defines...

of topological vector spaces, which include function spaces, inner product spaces, normed spaces, Hilbert spaces and Banach spaces. In this article, vectors...

translations which is equipped with an inner product. The action of translations makes the space an affine space, and this allows defining lines, planes...

function space. The inner product space is then called complete. A complete inner product space is a Hilbert space. The abstract state space is always...

well as physics, a linear operator A {\displaystyle A} acting on an inner product space is called positive-semidefinite (or non-negative) if, for every x...

a linear transformation is analogous to the null space of a matrix. If V is an inner product space, then the orthogonal complement to the kernel can...

Scalar-valued antilinear maps often arise when dealing with complex inner products and Hilbert spaces. A function is called antilinear or conjugate linear if it...

In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal unit vectors. A unit vector means that the vector has...

with infinitely many components in an inner product space, as in functional analysis. In an inner product space, the concept of perpendicularity is replaced...

. {\displaystyle \{{\vec {x}}\}^{\perp }.} For example, taking the inner product with a fixed function g ∈ L 2 ( [ − π , π ] ) {\displaystyle g\in L^{2}([-\pi...

tensor product. The tensor product allows Hilbert spaces to be collected into a symmetric monoidal category. Since Hilbert spaces have inner products, one...

an inner product ( ⋅ , ⋅ ) {\displaystyle (\cdot ,\cdot )} with antilinear first argument, which makes V {\displaystyle V} an inner product space. Then...

semidirect products Product of rings Ideal operations, for product of ideals Scalar multiplication Matrix multiplication Inner product, on an inner product space...

convexity is somewhere between an inner product space (all inner product spaces being strictly convex) and a general normed space in terms of structure. It also...

in an orthonormal basis over a real inner product space. The corresponding object for a complex inner product space is a Hermitian matrix with complex-valued...