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Given a vector bundle pi:E->M, its dual bundle is a vector bundle pi^*:E^*->M. The fiber bundle of E^* over a point p in M is the dual vector space to the fiber of E.

For every infinite loop space machine E, there is a natural equivalence of spectra between EX and Segal's spectrum BX.

Let A:D(A)->H and B:D(B)->H be linear operators from domains D(A) and D(B), respectively, into a Hilbert space H. It is said that B extends A if D(A) subset D(B) and if Bv=Av ...

For a function with 2 degrees of freedom, the 2-dimensional phase space that is accessible to the function or object is called its phase plane.

Every Lie algebra L is isomorphic to a subalgebra of some Lie algebra A^-, where the associative algebra A may be taken to be the linear operators over a vector space V.

In a complete metric space, a countable union of nowhere dense sets is said to be meager; the complement of such a set is a residual set.

A set in a Polish space is a Borel set iff it is both analytic and coanalytic. For subsets of w, a set is delta_1^1 iff it is "hyperarithmetic."

A topological space fulfilling the T_4-axiom: X fulfils the T1-separation axiom and is normal. In the terminology of Alexandroff and Hopf (1972), T_4-space are called Tietze ...

An abstract manifold is a manifold in the context of an abstract space with no particular embedding, or representation in mind. It is a topological space with an atlas of ...

A basis vector in an n-dimensional vector space is one of any chosen set of n vectors in the space forming a vector basis, i.e., having the property that every vector in the ...