inner product space oor Katalaans
inner product space
en
A vector space which is additionally equipped with an inner product.
Vertalings in die woordeboek Engels - Katalaans
espai prehilbertià
en
vector space with an additional structure called an inner product
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Complete inner product spaces are known as Hilbert spaces, in honor of David Hilbert.
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Thus suppose that H is an inner-product space.
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Its analogue over general inner product spaces is the Householder operator.
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Every finite-dimensional inner product space has an orthonormal basis, which may be obtained from an arbitrary basis using the Gram–Schmidt process.
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Thus the presence of an orthonormal basis reduces the study of a finite-dimensional inner product space to the study of Rn under dot product.
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The orthogonal complement generalizes to the annihilator, and gives a Galois connection on subsets of the inner product space, with associated closure operator the topological closure of the span.
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One requires a special kind of reversible function, namely a unitary mapping, that is, a linear transformation of a complex inner product space that preserves the Hermitian inner product.
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For a general inner product space V, an orthonormal basis can be used to define normalized orthogonal coordinates on V. Under these coordinates, the inner product becomes a dot product of vectors.
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One can recover the Fourier series version of Parseval's identity by letting H be the Hilbert space L2, and setting en = e−inx for n ∈ Z. More generally, Parseval's identity holds in any inner-product space, not just separable Hilbert spaces.
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Hilbert spaces: these have an inner product; even though these spaces may be infinite-dimensional, most geometrical reasoning familiar from finite dimensions can be carried out in them.
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Riemannian manifolds are manifolds whose tangent spaces are endowed with a suitable inner product.
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To see the inner product connection, consider a vector v in an n-dimensional real Euclidean space.
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Euclidean spaces: Rn or Cn with the topology induced by the standard inner product.
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Like a Riemannian metric, a Hermitian metric consists of a smoothly varying, positive definite inner product on the tangent bundle, which is Hermitian with respect to the complex structure on the tangent space at each point.
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