Definitions
from Wiktionary, Creative Commons Attribution/ShareAlike License
 n. Plural form of subspace.
Etymologies
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Examples

(An irreducible representation is one in which there are no subspaces invariant under the group action, apart from the null vector and the entire Hilbert space).

If the state space of an elementary system had relativistically invariant subspaces then it would be appropriate to associate these subspaces with elementary systems.

Put more technically, the state space of an elementary system must not contain any relativistically invariant subspaces, i.e., it must be the state space of an irreducible representation of the relevant invariance group.

Since a typical closed subspace has infinitely many complementary closed subspaces, this lattice is not distributive; however, it is orthocomplemented by the mapping

V over an involutive division ring is called a generalized Hilbert space if its lattice of closed subspaces

Ordered by setinclusion, the closed subspaces of H form a complete lattice, in which the meet (greatest lower bound) of a set of subspaces is their intersection, while their join (least upper bound) is the closed span of their union.

Axiom VII: The partially ordered set of all questions in quantum mechanics is isomorphic to the partially ordered set of all closed subspaces of a separable, infinite dimensional Hilbert space.

In view of the abovementioned oneone correspondence between closed subspaces and projections, we may impose upon the set

Such operators are in onetoone correspondence with the closed subspaces of H.

The Virasoro algebra acts on a fermionic Fock space and preserves certain subspaces generated by singular vectors; this means that the factor modules are well defined.
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