Given a vector space V over the field R of real numbers, a function is called sublinear if

The Hahn – Banach theorem states that if is a sublinear function, and is a linear functional on a linear subspace U ⊆ V which is dominated by on U,

For a function between metric spaces, it is equivalent to admit a modulus of continuity that is either concave, or subadditive, or uniformly continuous, or sublinear ( in the sense of growth ).

Thus, for a function between metric spaces it is equivalent to admit a modulus of continuity which is either concave, or subadditive, or uniformly continuous, or sublinear.

A sublinear modulus of continuity can easily found for any uniformly function which is a bounded perturbations of a Lipschitz function: if is a uniformly continuous function with modulus of continuity, and is a Lipschitz function with uniform distance from, then admits the sublinear module of continuity Conversely, at least for real-valued functions, any bounded, uniformly continuous perturbation of a Lipschitz function is a special uniformly continuous function ; indeed more is true as shown below.

Note that as an immediate consequence, any uniformly continuous function on a convex subset of a normed space has a sublinear growth: there are constants and such that for all.

In functional analysis the name Banach functional is used for sublinear function, especially when formulating Hahn – Banach theorem.

In computer science, a function is called sublinear if in asymptotic notation ( Notice the small ).

Every seminorm on V ( in particular, every norm on V ) is sublinear.

Several important space complexity classes are sublinear, that is, smaller than the size of the input.

* is sublinear, that is, there are constants and such that for all ;

One-way functions are necessary, but not known to be sufficient, for nontrivial ( i. e., with sublinear communication ) single database computationally private information retrieval.

Other sublinear functions can be useful as well, especially Minkowski functionals of convex sets.

** Search for a regular expression P in time expected sublinear in.

In this section we mainly deal with moduli of continuity that are concave, or subadditive, or uniformly continuous, or sublinear.

Even more strongly, for any fixed k, only a sublinear number of values of n need more than two terms in their Egyptian fraction expansions.

The generalized version of the conjecture is equivalent to the statement that the number of unexpandable fractions is not just sublinear but bounded.

It can find all the primes up to N in time O ( N ), while the sieve of Atkin and most wheel sieves run in sublinear time O ( N / log log N ).

Moreover, whereas in Interstate Commerce Commission parlance `` variable cost '' means a cost deemed to vary in direct proportion to changes in rate of output, in the type of analysis now under review `` variable cost '' has been used more broadly, so as to cover costs which, while a function of some one variable ( such as output of energy, or number of customers ), are not necessarily a linear function.

In linear algebra, a bilinear transformation is a binary function where the sets X, Y, and Z are all vector spaces and the derived functions f < sup > x </ sup > and f < sub > y </ sub > are all linear transformations.

A bilinear transformation, like any binary function, can be interpreted as a function from X × Y to Z, but this function in general won't be linear.

The tensor product X ⊗ Y from X and Y is a K-vector space Z with a bilinear function T: X × Y → Z which has the following universal property: If T ′: X × Y → Z ′ is any bilinear function into a K-vector space Z ′, then only one linear function f: Z → Z ′ with exists.

In mathematics, a bilinear operator is a function combining elements of two vector spaces to yield an element of a third vector space that is linear in each of its arguments.

The character function is also used to display the timecode on the preview monitors in linear editing suites.

The bilinear transform is a special case of a conformal mapping ( namely, the Möbius transformation ), often used to convert a transfer function of a linear, time-invariant ( LTI ) filter in the continuous-time domain ( often called an analog filter ) to a transfer function of a linear, shift-invariant filter in the discrete-time domain ( often called a digital filter although there are analog filters constructed with switched capacitors that are discrete-time filters ).

The total derivative is a linear transformation that captures how the function changes in all directions.

In the language of linear transformations, D < sub > a </ sub >( g ) is the function which scales a vector by a factor of g ′( a ) and D < sub > g ( a )</ sub >( f ) is the function which scales a vector by a factor of f ′( g ( a )).

The chain rule says that the composite of these two linear transformations is the linear transformation, and therefore it is the function that scales a vector by f ′( g ( a )) g ′( a ).

Given a function f ∈ I < sub > x </ sub > ( a smooth function vanishing at x ) we can form the linear functional df < sub > x </ sub > as above.

If the resulting linear differential equations have constant coefficients one can take their Laplace transform to obtain a transfer function.

The transfer function is a mathematical representation, in terms of spatial or temporal frequency, of the relation between the input and output of a linear time-invariant solution of the nonlinear differential equations describing the system.

* Natural Banach function algebra: A uniform algebra whose all characters are evaluations at points of X.

Several elementary functions which are defined via power series may be defined in any unital Banach algebra ; examples include the exponential function and the trigonometric functions, and more generally any entire function.

where is the Gelfand representation of x defined as follows: is the continuous function from Δ ( A ) to C given by The spectrum of in the formula above, is the spectrum as element of the algebra C ( Δ ( A )) of complex continuous functions on the compact space Δ ( A ).

In his 1799 doctorate in absentia, A new proof of the theorem that every integral rational algebraic function of one variable can be resolved into real factors of the first or second degree, Gauss proved the fundamental theorem of algebra which states that every non-constant single-variable polynomial with complex coefficients has at least one complex root.

* 1799: Doctoral dissertation on the Fundamental theorem of algebra, with the title: Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse (" New proof of the theorem that every integral algebraic function of one variable can be resolved into real factors ( i. e., polynomials ) of the first or second degree ")

It is often convenient to express the theory using the algebra of random variables: thus if X is used to denote a random variable corresponding to the observed data, the estimator ( itself treated as a random variable ) is symbolised as a function of that random variable,.

In mathematics, more specifically in abstract algebra and ring theory, a Euclidean domain ( also called a Euclidean ring ) is a ring that can be endowed with a certain structure – namely a Euclidean function, to be described in detail below – which allows a suitable generalization of the Euclidean division of the integers.

Most algebra texts require a Euclidean function to have the following additional property:

The mathematical definition of an elementary function, or a function in elementary form, is considered in the context of differential algebra.

In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations.

This is called the exponential map, and it maps the Lie algebra into the Lie group G. It provides a diffeomorphism between a neighborhood of 0 in and a neighborhood of e in G. This exponential map is a generalization of the exponential function for real numbers ( because R is the Lie algebra of the Lie group of positive real numbers with multiplication ), for complex numbers ( because C is the Lie algebra of the Lie group of non-zero complex numbers with multiplication ) and for matrices ( because M < sub > n </ sub >( R ) with the regular commutator is the Lie algebra of the Lie group GL < sub > n </ sub >( R ) of all invertible matrices ).

If R is commutative, then one can associate to every polynomial P in R, a polynomial function f with domain and range equal to R ( more generally one can take domain and range to be the same unital associative algebra over R ).

In ring theory or abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication.

* Banach function algebra

This also was a great triumph for the new equation, as it traced the mysterious i that appears in it, and the necessity of a complex wave function, back to the geometry of space-time through the Dirac algebra.

In particular, when a function f: R → R is said to be Lebesgue measurable what is actually meant is that is a measurable function — that is, the domain and range represent different σ-algebras on the same underlying set ( here is the sigma algebra of Lebesgue measurable sets, and is the Borel algebra on R ).