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A covariant tensor of rank 1, more commonly called a one-form (or "bra").

In abstract topology, a machine is method for producing infinite loop spaces and spectra. In automata theory, an abstract machine that is implemented in hardware is simply ...

Let H be a Hilbert space and M a closed subspace of H. Corresponding to any vector x in H, there is a unique vector m_0 in M such that |x-m_0|<=|x-m| for all m in M. ...

Let theta be the angle between two vectors. If 0<theta<pi, the vectors are positively oriented. If pi<theta<2pi, the vectors are negatively oriented. Two vectors in the plane ...

A Hilbert basis for the vector space of square summable sequences (a_n)=a_1, a_2, ... is given by the standard basis e_i, where e_i=delta_(in), with delta_(in) the Kronecker ...

Let V=R^k be a k-dimensional vector space over R, let S subset V, and let W={w in V:w·n^^=0} be a subspace of V of dimension k-1, where n^^ is a unit normal vector of W. Then ...

Exterior algebra is the algebra of the wedge product, also called an alternating algebra or Grassmann algebra. The study of exterior algebra is also called Ausdehnungslehre ...

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 ...

The tensor product between modules A and B is a more general notion than the vector space tensor product. In this case, we replace "scalars" by a ring R. The familiar ...

The coordinates representing any point of an n-dimensional affine space A by an n-tuple of real numbers, thus establishing a one-to-one correspondence between A and R^n. If V ...