For vector spaces over non-algebraically closed fields, we still need to find some substitute for characteristic values and vectors.

For example an absolute value is also defined for the complex numbers, the quaternions, ordered rings, fields and vector spaces.

For example, upon rolling a 5-5 that player may move up to four separate checkers forward five spaces each.

For complete metric spaces this is equivalent to compactness.

For bassists with smaller hands, the large spaces between pitches may present a significant challenge, especially in the lowest range, where the spaces between notes are largest.

For example, if a child falls into a well and struggles to get out, he or she may develop a fear of wells, heights ( acrophobia ), enclosed spaces ( claustrophobia ), or water ( aquaphobia ).

For path-connected spaces, therefore, we can write π < sub > 1 </ sub >( X ) instead of π < sub > 1 </ sub >( X, x < sub > 0 </ sub >) without ambiguity whenever we care about the isomorphism class only.

For example, a bijective linear map is an isomorphism between vector spaces, and a bijective continuous function whose inverse is also continuous is an isomorphism between topological spaces, called a homeomorphism.

For example, they are Hausdorff paracompact spaces ( and hence normal and Tychonoff ) and first-countable.

For measure spaces that are also topological spaces various compatibility conditions can be placed for the measure and the topology.

For a language like English, this is fairly trivial, since words are usually separated by spaces.

For locally compact spaces an integration theory is then recovered.

For natural lighting of interior spaces by admitting sunlight, see Daylighting.

For a given quadratic form, several different spaces of spinors with extra properties may exist.

For a brief period in fact, complete normed linear spaces where referred to as " Banach-Wiener " spaces in mathematical literature, based on terminology introduced by Wiener himself.

For instance, any isometry ( distance-preserving map ) between metric spaces is uniformly continuous.

For every vector space there exists a basis ( if one assumes the axiom of choice ), and all bases of a vector space have equal cardinality ( see dimension theorem for vector spaces ); as a result the dimension of a vector space is uniquely defined.

For example, algebraic topology makes use of Eilenberg – MacLane spaces which are spaces with prescribed homotopy groups.

For example, a topological group is just a group in the category of topological spaces.

For example, a corollary of the lemma is that normal T < sub > 1 </ sub > spaces are Tychonoff.

0 implies that any function f that is holomorphic on the simply connected region U is also integrable on U. ( For a path γ from z < sub > 0 </ sub > to z lying entirely in U, define

For every Riemann surface, a meromorphic function is the same as a holomorphic function that maps to the Riemann sphere and which is not constant ∞.

i. e., the dimension of the space of functions that are holomorphic everywhere except at P where the function is allowed to have a pole of order at most n. For n = 0, the functions are thus required to be entire, i. e., holomorphic on the whole surface X.

For discrete-time dynamical systems the orbits are sequences, for real dynamical systems the orbits are curves and for holomorphic dynamical systems the orbits are Riemann surfaces.

For example, suppose that ƒ < sub > 1 </ sub >, ƒ < sub > 2 </ sub >, ... is a sequence of holomorphic functions, converging uniformly to a continuous function ƒ on an open disc.

For a closed surface the invariant of the theory is the number of pseudo holomorphic maps in the sense of Gromov ( they are ordinary holomorphic maps if is a Kaehler manifold ).

For any point P ∈ X and Q = ƒ ( P ) ∈ Y, there are holomorphic local coordinates z for X near P and w for Y near Q in terms of which the function ƒ ( z ) is given by

For varieties of higher dimension the construction of the Jacobian variety as a quotient of the space of holomorphic 1-forms generalizes to give the Albanese variety, but in general this need not be isomorphic to the Picard variety.

For example, the Whitney embedding theorem tells us that every smooth manifold can be embedded as a smooth submanifold of R < sup > n </ sup >, whereas it is " rare " for a complex manifold to have a holomorphic embedding into C < sup > n </ sup >.

For instance, every holomorphic function is representable as power series in every open disc in its domain of definition, and is therefore analytic.

For values with imaginary part between-1 and 1, the arctangent function is holomorphic, and it is easily seen that the series converges uniformly on compact sets in the region with imaginary part between-1 / 2 and 1 / 2, leading to a holomorphic function on this domain.

For real analytic functions, unlike for complex analytic ( that is, holomorphic ) functions, these statements fail to hold.

For each p, a holomorphic p-form is a holomorphic section of the bundle Ω < sup > p, 0 </ sup >.

For an appropriate choice of almost complex structure, punctured holomorphic curves ( of finite energy ) in it have cylindrical ends asymptotic to the loops in the mapping torus corresponding to fixed points of the symplectomorphism.

For the interpretation of these significant investigations, it should be remembered that reciprocal relations exist in the hypothalamus with respect to autonomic and somatic functions which are closely associated with the emotions.

For many years he had provided music and entertainment for functions throughout the Northwest.

For the Greeks, Apollo was all the Gods in one and through the centuries he acquired different functions which could originate from different gods.

For liturgical functions other than the Mass the bishop typically wears the cope.

For example, one cannot define all the trigonometric functions in a Banach algebra without identity.

For non-integer α, the functions J < sub > α </ sub >( x ) and J < sub >− α </ sub >( x ) are linearly independent, and are therefore the two solutions of the differential equation.

For the spherical Bessel functions the orthogonality relation is:

For instance, NASA has extremely rigorous software testing procedures for many operating systems and communication functions.

For a certain class of Green functions coming from solutions of integral equations, Schmidt had shown that a property analogous to the Arzelà – Ascoli theorem held in the sense of mean convergence — or convergence in what would later be dubbed a Hilbert space.

For those priests over 80 who became cardinal-deacons and were not ordained to the episcopacy, this is the highest position they can normally attain in the Church hierarchy ( though all cardinals rank above bishops in rank and order of precedence, those cardinals who are not bishops do not have the right to perform the functions reserved solely to bishops, such as ordination ).

For some distributed parameter systems the vectors may be infinite-dimensional ( typically functions ).

For instance, instead of a hardware multiplier, a calculator might implement floating point mathematics with code in ROM, and compute trigonometric functions with the CORDIC algorithm because CORDIC does not require hardware floating-point.

For example: clairsentience is one of the six human special functions mentioned or recorded in Buddhism.

For the same reason, filter functions whose critical response is at lower frequencies ( compared to the sampling frequency 1 / T ) require a higher order, more computationally intensive FIR filter.

For polynomials of degree four and smaller there are explicit formulae for the roots ( the formulae are elementary functions ).

For the case of a carrier modulated by a single sine wave, the resulting frequency spectrum can be calculated using Bessel functions of the first kind, as a function of the sideband number and the modulation index.

For instance, by assigning to every open set U the associative algebra of real-valued continuous functions on U, one obtains a presheaf of algebras on X.

For periodic functions, both the Fourier transform and the DTFT comprise only a discrete set of frequency components ( Fourier series ), and the transforms diverge at those frequencies.

Direction of causal flow is upward. For a multivariate function, functional decomposition generally refers to a process of identifying a set of functions such that

For crimes that are not committed in the exercise of his functions, the President of the Republic answers before the common courts, once his term of office has ended.

For example, operation-preserving functions exist between the set of real numbers with addition and the set of positive real numbers with multiplication.

For most types of hashing functions the choice of the function depends strongly on the nature of the input data, and their probability distribution in the intended application.

For instance, the inverse of the sine function is typically called the arcsine function, written as arcsin, which is, like sin, conventionally denoted in roman type and not in italics ( note that software libraries of mathematical functions often use the name < tt > asin </ tt >):

For example, scientists developing a new viral drug to treat an infection with a pathogenic virus ( e. g. HIV-1 ) may find that a candidate drug functions to prevent viral replication in an in vitro setting ( typically cell culture ).

For this purpose the Internet Protocol defines an addressing system that has two functions: identifying hosts and providing a logical location service.