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Given a topological vector space ( X, τ ) over a field F, S is called bounded if for every neighborhood N of the zero vector there exists a scalar α so that
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Given and topological
* Given any topological space X, the continuous real-or complex-valued functions on X form a real or complex unitary associative algebra ; here the functions are added and multiplied pointwise.
Given a point x in a topological space, let N < sub > x </ sub > denote the set of all neighbourhoods containing x.
Given an arbitrary topological space ( X, τ ) there is a universal way of associating a completely regular space with ( X, τ ).
Given a topological space X, denote F the set of filters on X, x ∈ X a point, V ( x ) ∈ F the neighborhood filter of x, A ∈ F a particular filter and the set of filters finer than A and that converge to x.
Given a topological space X, a base for the closed sets of X is a family of closed sets F such that any closed set A is an intersection of members of F.
Given any topological space X, the zero sets form the base for the closed sets of some topology on X.
Given a topological space X, a subset A of X is meagre if it can be expressed as the union of countably many nowhere dense subsets of X.
Given a point x of a topological space X, and two maps f, g: X → Y ( where Y is any set ), then f and g define the same germ at x if there is a neighbourhood U of x such that restricted to U, f and g are equal ;
Given any unital ring R, the set of singular n-simplices on a topological space can be taken to be the generators of a free R-module.
Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action.
Given an action of a group G on a topological space X by homeomorphisms, a fundamental domain ( also called fundamental region ) for this action is a set D of representatives for the orbits.
Given a compact topological space X, the topological K-theory K < sup > top </ sup >( X ) of ( real ) vector bundles over X coincides with K < sub > 0 </ sub > of the ring of continuous real-valued functions on X.
Given a topological group G, the Bohr compactification of G is a compact Hausdorff topological group Bohr ( G ) and a continuous homomorphism
Given any topological space X we can define a ( possibly ) finer topology on X which is compactly generated.
Given a field of sets the complexes form a base for a topology, we denote the corresponding topological space by.
Given a topological space the clopen sets trivially form a topological field of sets as each clopen set is its own interior and closure.
Given an interior algebra we can form the Stone representation of its underlying Boolean algebra and then extend this to a topological field of sets by taking the topology generated by the complexes corresponding to the open elements of the interior algebra ( which form a base for a topology ).
Given and vector
Given any vector space V over a field F, the dual space V * is defined as the set of all linear maps ( linear functionals ).
Given a basis of a vector space, every element of the vector space can be expressed uniquely as a finite linear combination of basis vectors.
Given a vector space V and a quadratic form g an explicit matrix representation of the Clifford algebra can be defined as follows.
Given a vector v in R < sup > n </ sup > one defines the directional derivative of a smooth map ƒ: R < sup > n </ sup >→ R at a point x by
Given any vector space V over K we can construct the tensor algebra T ( V ) of V. The tensor algebra is characterized by the fact:
Given that this is a plane wave, each vector represents the magnitude and direction of the electric field for an entire plane that is perpendicular to the axis.
Given a normalized light vector l ( pointing from the light source toward the surface ) and a normalized plane normal vector n, one can work out the normalized reflected and refracted rays:
Given the dimensions of the ellipsoid, the conversion from lat / lon / height-above-ellipsoid coordinates to X-Y-Z is straightforward — calculate the X-Y-Z for the given lat-lon on the surface of the ellipsoid and add the X-Y-Z vector that is perpendicular to the ellipsoid there and has length equal to the point's height above the ellipsoid.
Given a vector space V over a field K, the span of a set S ( not necessarily finite ) is defined to be the intersection W of all subspaces of V which contain S. W is referred to as the subspace spanned by S, or by the vectors in S. Conversely, S is called a spanning set of W.
Given a subset S in R < sup > n </ sup >, a vector field is represented by a vector-valued function V: S → R < sup > n </ sup > in standard Cartesian coordinates ( x < sub > 1 </ sub >, ..., x < sub > n </ sub >).
Given two C < sup > k </ sup >- vector fields V, W defined on S and a real valued C < sup > k </ sup >- function f defined on S, the two operations scalar multiplication and vector addition
Given a differentiable manifold M, a vector field on M is an assignment of a tangent vector to each point in M. More precisely, a vector field F is a mapping from M into the tangent bundle TM so that is the identity mapping
Given a vector x ∈ V and y * ∈ W *, then the tensor product y * ⊗ x corresponds to the map A: W → V given by
Given ω = ( xθ, yθ, zθ ), with v = ( x, y, z ) a unit vector, the correct skew-symmetric matrix form of ω is
Given two column vectors, their dot product can also be obtained by multiplying the transpose of one vector with the other vector and extracting the unique coefficient of the resulting 1 × 1 matrix.
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