To install click the Add extension button. That's it.

The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time.

4,5
Kelly Slayton
Congratulations on this excellent venture… what a great idea!
Alexander Grigorievskiy
I use WIKI 2 every day and almost forgot how the original Wikipedia looks like.
Live Statistics
English Articles
Improved in 24 Hours
Added in 24 Hours
What we do. Every page goes through several hundred of perfecting techniques; in live mode. Quite the same Wikipedia. Just better.
.
Leo
Newton
Brights
Milds

From Wikipedia, the free encyclopedia

The perimeter of the square is the set of points in 2 where the sup norm equals a fixed positive constant. For example, points (2, 0), (2, 1), and (2, 2) lie along the perimeter of a square and belong to the set of vectors whose sup norm is 2.

In mathematical analysis, the uniform norm (or sup norm) assigns to real- or complex-valued bounded functions defined on a set the non-negative number

This norm is also called the supremum norm, the Chebyshev norm, the infinity norm, or, when the supremum is in fact the maximum, the max norm. The name "uniform norm" derives from the fact that a sequence of functions converges to under the metric derived from the uniform norm if and only if converges to uniformly.[1]

If is a continuous function on a closed and bounded interval, or more generally a compact set, then it is bounded and the supremum in the above definition is attained by the Weierstrass extreme value theorem, so we can replace the supremum by the maximum. In this case, the norm is also called the maximum norm. In particular, if is some vector such that in finite dimensional coordinate space, it takes the form:

This is called the -norm.

YouTube Encyclopedic

  • 1/3
    Views:
    6 319
    50 038
    1 928
  • Intro Real Analysis, Lec 35, Sup Norm and Metric on C[a,b], Sequence Space, Open & Closed Sets
  • Visualizing norms as a unit circle
  • Functional Analysis - Part 14 - Example Operator Norm

Transcription

Metric and topology

The metric generated by this norm is called the Chebyshev metric, after Pafnuty Chebyshev, who was first to systematically study it.

If we allow unbounded functions, this formula does not yield a norm or metric in a strict sense, although the obtained so-called extended metric still allows one to define a topology on the function space in question.

The binary function

is then a metric on the space of all bounded functions (and, obviously, any of its subsets) on a particular domain. A sequence converges uniformly to a function if and only if

We can define closed sets and closures of sets with respect to this metric topology; closed sets in the uniform norm are sometimes called uniformly closed and closures uniform closures. The uniform closure of a set of functions A is the space of all functions that can be approximated by a sequence of uniformly-converging functions on For instance, one restatement of the Stone–Weierstrass theorem is that the set of all continuous functions on is the uniform closure of the set of polynomials on

For complex continuous functions over a compact space, this turns it into a C* algebra.

Properties

The set of vectors whose infinity norm is a given constant, forms the surface of a hypercube with edge length 

The reason for the subscript “” is that whenever is continuous and for some , then

where
where is the domain of ; the integral amounts to a sum if is a discrete set (see p-norm).

See also

References

  1. ^ Rudin, Walter (1964). Principles of Mathematical Analysis. New York: McGraw-Hill. pp. 151. ISBN 0-07-054235-X.
This page was last edited on 23 January 2024, at 08:18
Basis of this page is in Wikipedia. Text is available under the CC BY-SA 3.0 Unported License. Non-text media are available under their specified licenses. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc. WIKI 2 is an independent company and has no affiliation with Wikimedia Foundation.