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Let A be an involutive algebra over the field C of complex numbers with involution xi|->xi^♭. Then A is a right Hilbert algebra if A has an inner product <·,·> satisfying: 1. ...

Let M be a regular surface with v_(p),w_(p) points in the tangent space M_(p) of M. For M in R^3, the second fundamental form is the symmetric bilinear form on the tangent ...

Let A be a C^*-algebra. An element a in A is called self-adjoint if a^*=a. For example, the real functions of the C^*-algebra of C([a,b]) of continuous complex-valued ...

A mathematical object defined for a set and a binary operator in which the multiplication operation is associative. No other restrictions are placed on a semigroup; thus a ...

A strong Riemannian metric on a smooth manifold M is a (0,2) tensor field g which is both a strong pseudo-Riemannian metric and positive definite. In a very precise way, the ...

A left Hilbert Algebra A whose involution is an antilinear isometry is called a unimodular Hilbert algebra. The involution is usually denoted xi|->xi^*.

Let A be a unital C^*-algebra. An element u in A is called unitary if u^*u=uu^*=1. For example, for each self-adjoint element a in A, the element ...

A W^*-algebra is a C-*-algebra A for which there is a Banach space A_* such that its dual is A. Then the space A_* is uniquely defined and is called the pre-dual of A. Every ...

The Weingarten equations express the derivatives of the normal vector to a surface using derivatives of the position vector. Let x:U->R^3 be a regular patch, then the shape ...

The Monty Hall problem is named for its similarity to the Let's Make a Deal television game show hosted by Monty Hall. The problem is stated as follows. Assume that a room is ...