Subspaces in a sentence

Angles between subspaces The definition of the angle between one-dimensional subspaces and given by : in a Hilbert space can be extended to subspaces of any finite dimensions.

For example, both the range and kernel of a linear mapping are subspaces, and are thus often called the range space and the nullspace ; these are important examples of subspaces.

Operations and relations on subspaces Inclusion The set-theoretical inclusion binary relation specifies a partial order on the set of all subspaces (of any dimension).

That left a sparse hodgepodge of subspaces that they still had to deal with.

They then applied a modified version of a fundamentally random process called the Rödl nibble to cover most of the remaining subspaces.

A Banach space isomorphic to all its infinite-dimensional closed subspaces is isomorphic to a separable Hilbert space.

An infinite-dimensional Banach space X is said to be homogeneous if it is isomorphic to all its infinite-dimensional closed subspaces.

Assume the following time average exists: : For quantum mechanical systems, an important assumption made in the quantum logic approach to quantum mechanics is the identification of yes-no questions to the lattice of closed subspaces of a Hilbert space.

Closed linear spans are important when dealing with closed linear subspaces (which are themselves highly important, consider Riesz's lemma ).

Earlier, Menger and Birkhoff had axiomatized complex projective geometry in terms of the properties of its lattice of linear subspaces.

Hermitian vector spaces and spinors If the vector space V has extra structure that provides a decomposition of its complexification into two maximal isotropic subspaces, then the definition of spinors (by either method) becomes natural.

If is the norm (usually denoted by ) defined in the sequence space ℓ ∞ of all bounded sequences (which matches the maximum of distances measured on projections into the base subspaces), and is lower triangular non-singular (i.

In this case, N and N′ are complementary subspaces.

In this way, the spinor representations are identified with certain subspaces of the Clifford algebra itself.

It includes the study of lines, planes, and subspaces, but is also concerned with properties common to all vector spaces.

It is interested in all the ways that this is possible, and it does so by finding subspaces invariant under all transformations of the algebra.

Rays (one-dimensional subspaces) in H are associated with states of the system.

Springer Verlag Working within GA, Euclidean space is embedded projectively in the CGA via the identification of Euclidean points with 1D subspaces in the 4D null cone of the 5D CGA vector subspace, and adding a point at infinity.

Subspaces and quotient spaces main A line passing through the origin (blue, thick) in R 3 is a linear subspace.

Subspaces of V are vector spaces (over the same field) in their own right.