usual inner product - vertaling naar russisch
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usual inner product - vertaling naar russisch

REAL OR COMPLEX VECTOR SPACE WITH AN ADDITIONAL STRUCTURE CALLED AN INNER PRODUCT
Linear Algebra/Inner Product Space; Inner product; Inner-product space; Inner products; Pre-Hilbert space; Inner Product Space; Inner product spaces; Scalar product space; Prehilbert space; Unitary space; Inner-product; Orthogonal vector; Orthogonal vectors; Inner product of vectors; Hermitian inner product; Orthogonal vectors (inner product space)
  • Geometric interpretation of the angle between two vectors defined using an inner product
  • Scalar product spaces, over any field, have "scalar products" that are symmetrical and linear in the first argument. Hermitian product spaces are restricted to the field of complex numbers and have "Hermitian products" that are conjugate-symmetrical and linear in the first argument. Inner product spaces may be defined over any field, having "inner products" that are linear in the first argument, conjugate-symmetrical, and positive-definite. Unlike inner products, scalar products and Hermitian products need not be positive-definite.

usual inner product      
стандартное внутреннее произведение
inner product         

физика

проигрыш энергии скалярный

внутреннее произведение

unitary space         

математика

унитарное пространство

Definitie

inner product
<mathematics> In linear algebra, any linear map from a vector space to its dual defines a product on the vector space: for u, v in V and linear g: V -> V' we have gu in V' so (gu): V -> scalars, whence (gu)(v) is a scalar, known as the inner product of u and v under g. If the value of this scalar is unchanged under interchange of u and v (i.e. (gu)(v) = (gv)(u)), we say the inner product, g, is symmetric. Attention is seldom paid to any other kind of inner product. An inner product, g: V -> V', is said to be positive definite iff, for all non-zero v in V, (gv)v > 0; likewise negative definite iff all such (gv)v < 0; positive semi-definite or non-negative definite iff all such (gv)v >= 0; negative semi-definite or non-positive definite iff all such (gv)v <= 0. Outside relativity, attention is seldom paid to any but positive definite inner products. Where only one inner product enters into discussion, it is generally elided in favour of some piece of syntactic sugar, like a big dot between the two vectors, and practitioners don't take much effort to distinguish between vectors and their duals. (1997-03-16)

Wikipedia

Inner product space

In mathematics, 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. 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 } . Inner products allow formal definitions of intuitive geometric notions, such as lengths, angles, and orthogonality (zero inner product) of vectors. Inner product spaces generalize Euclidean vector spaces, in which the inner product is the dot product or scalar product of Cartesian coordinates. Inner product spaces of infinite dimension are widely used in functional analysis. Inner product spaces over the field of complex numbers are sometimes referred to as unitary spaces. The first usage of the concept of a vector space with an inner product is due to Giuseppe Peano, in 1898.

An inner product naturally induces an associated norm, (denoted | x | {\displaystyle |x|} and | y | {\displaystyle |y|} in the picture); so, every inner product space is a normed vector space. If this normed space is also complete (that is, a Banach space) then the inner product space is a Hilbert space. If an inner product space H is not a Hilbert space, it can be extended by completion to a Hilbert space H ¯ . {\displaystyle {\overline {H}}.} This means that H {\displaystyle H} is a linear subspace of H ¯ , {\displaystyle {\overline {H}},} the inner product of H {\displaystyle H} is the restriction of that of H ¯ , {\displaystyle {\overline {H}},} and H {\displaystyle H} is dense in H ¯ {\displaystyle {\overline {H}}} for the topology defined by the norm.

Vertaling van &#39usual inner product&#39 naar Russisch