With no explanation, label text_A→text_B with either "not_related" or "related".
text_A: Holiday: A Soldier Is Never Off Duty is Indian.
text_B: In functional analysis , a branch of mathematics , nest algebras are a class of operator algebras that generalise the upper-triangular matrix algebras to a Hilbert space context .. functional analysis. functional analysis. upper-triangular matrix. upper-triangular matrix. Hilbert space. Hilbert space. They were introduced by and have many interesting properties .. They are non-selfadjoint algebras , are closed in the weak operator topology and are reflexive .. selfadjoint. selfadjoint. closed. closure ( mathematics ). weak operator topology. weak operator topology. reflexive. reflexive operator algebra. Nest algebras are among the simplest examples of commutative subspace lattice algebras .. subspace. Linear subspace. Indeed , they are formally defined as the algebra of bounded operators leaving invariant each subspace contained in a subspace nest , that is , a set of subspaces which is totally ordered by inclusion and is also a complete lattice .. invariant. Invariant ( mathematics ). subspace. Linear subspace. subspace nest. subspace nest. totally ordered. total order. inclusion. subset. complete lattice. complete lattice. Since the orthogonal projections corresponding to the subspaces in a nest commute , nests are commutative subspace lattices .. subspace. Linear subspace. commute. commutative operation. By way of an example , let us apply this definition to recover the finite-dimensional upper-triangular matrices .. Let us work in the - dimensional complex vector space , and let be the standard basis .. complex. complex number. vector space. vector space. standard basis. standard basis. For , let be the - dimensional subspace of spanned by the first basis vectors .. subspace. Linear subspace. Let. then N is a subspace nest , and the corresponding nest algebra of n × n complex matrices M leaving each subspace in N invariant that is , satisfying for each S in N -- is precisely the set of upper-triangular matrices .. invariant. Invariant ( mathematics ). subspace. Linear subspace. subspace nest. subspace nest. complex. complex number. If we omit one or more of the subspaces Sj from N then the corresponding nest algebra consists of block upper-triangular matrices .
not_related.