Pointwise in a sentence

The sequence with converges pointwise but not uniformly: : In this example one can easily see that pointwise convergence does not preserve differentiability or continuity.

Equipped with the topology of pointwise convergence on A (i.

Function spaces Functions from any fixed set Ω to a field F also form vector spaces, by performing addition and scalar multiplication pointwise.

If we use complex-valued functions, the space L ∞ is a commutative C*-algebra with pointwise multiplication and conjugation.

Indeed, it coincides with the topology of pointwise convergence of linear functionals.

In symbols, U is pointwise finite if and only if, for any x in X, the set : : is finite.

It need not hold in a pointwise sense, even when f is a continuous function.

Local continuity versus global uniform continuity Continuity itself is a local (more precisely, pointwise) property of a function—that is, a function f is continuous, or not, at a particular point.

Normal set of chess pieces move edgewise or pointwise.

Series of functions main A series of real- or complex-valued functions : converges pointwise on a set E, if the series converges for each x in E as an ordinary series of real or complex numbers.

The formal definition and the distinction between pointwise continuity and uniform continuity were first given by Bolzano in the 1830s but the work wasn't published until the 1930s.

The operations are pointwise addition and multiplication of functions.

The set of all such formal power series is denoted R«X I », and it is given a ring structure by defining addition pointwise : and multiplication by : where · denotes concatenation of words.

The solution to this conundrum, carried out first in Bishop's 1967 book, is to consider only functions which are written as the pointwise limit of continuous functions (with known modulus of continuity), with information about the rate of convergence.

The space of complex-valued continuous functions on X that vanish at infinity (defined in the article on local compactness ) form a commutative C*-algebra under pointwise multiplication and addition.

This choice of harmonics enjoys some of the useful properties of the classical Fourier transform in terms of carrying convolutions to pointwise products, or otherwise showing a certain understanding of the underlying group structure.

This generalizes the Fourier transform to all spaces of the form L 2 (G), where G is a compact group, in such a way that the Fourier transform carries convolutions to pointwise products.

This theorem is important, since pointwise convergence of continuous functions is not enough to guarantee continuity of the limit function as the image illustrates.

To define them, we first need to extend the list of terms above: A topological space is: * metacompact if every open cover has an open pointwise finite refinement.

Uniform convergence is also guaranteed if is a compact interval and is an equicontinuous sequence that converges pointwise.