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In [[functional analysis]], it is often convenient to define a [[linear transformation]] on a [[complete space|complete]], [[normed vector space]] <math>X</math> by first defining a linear transformation <math>\mathsf{T}</math> on a [[dense set|dense]] [[subset]] of <math>X</math> and then extending <math>\mathsf{T}</math> to the whole space via the theorem below. The resulting extension remains [[linearity|linear]] and [[bounded operator|bounded]] (thus [[continuous function|continuous]]).
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This procedure is known as '''continuous linear extension'''.
 
==Theorem==
 
Every bounded linear transformation <math>\mathsf{T}</math> from a normed vector space <math>X</math> to a complete, normed vector space <math>Y</math> can be uniquely extended to a bounded linear transformation <math>\tilde{\mathsf{T}}</math> from the [[complete space#Completion|completion]] of  <math>X</math> to <math>Y</math>. In addition, the [[operator norm]] of <math>\mathsf{T}</math> is <math>c</math> [[iff]] the norm of <math>\tilde{\mathsf{T}}</math> is <math>c</math>.
 
This theorem is sometimes called the B L T theorem, where B L T stands for ''bounded linear transformation''.
 
==Application==
 
Consider, for instance, the definition of the [[Riemann integral]]. A [[step function]] on a [[closure (mathematics)|closed]] [[interval (mathematics)|interval]] <math>[a,b]</math>  is a function of the form: <math>f\equiv r_1 \mathit{1}_{[a,x_1)}+r_2 \mathit{1}_{[x_1,x_2)} + \cdots + r_n \mathit{1}_{[x_{n-1},b]}</math>
where <math>r_1, \ldots, r_n</math> are real numbers, <math>a=x_0<x_1<\ldots <x_{n-1}<x_n=b</math>, and <math>\mathit{1}_S</math> denotes the [[indicator function]] of the set <math>S</math>. The space of all step functions on <math>[a,b]</math>, normed by the <math>L^\infty</math> norm (see [[Lp space]]), is a normed vector space which we denote by <math>\mathcal{S}</math>. Define the integral of a step function by: <math>\mathsf{I} \left(\sum_{i=1}^n r_i \mathit{1}_{ [x_{i-1},x_i)}\right) = \sum_{i=1}^n r_i (x_i-x_{i-1})</math>.
<math>\mathsf{I}</math> as a function is a bounded linear transformation from <math>\mathcal{S}</math> into <math>\mathbb{R}</math>.<ref>Here, <math>\mathbb{R}</math> is also a normed vector space; <math>\mathbb{R}</math> is a vector space because it satisfies all of the [[vector space#Formal_definition|vector space axioms]] and is normed by the [[absolute value|absolute value function]].</ref>
 
Let <math>\mathcal{PC}</math> denote the space of bounded, [[piecewise]] continuous functions on <math>[a,b]</math> that are continuous from the right, along with the <math>L^\infty</math> norm. The space <math>\mathcal{S}</math> is dense in <math>\mathcal{PC}</math>, so we can apply the B.L.T. theorem to extend the linear transformation <math>\mathsf{I}</math> to a bounded linear transformation <math>\tilde{\mathsf{I}}</math> from <math>\mathcal{PC}</math> to <math>\mathbb{R}</math>. This defines the Riemann integral of all functions in <math>\mathcal{PC}</math>; for every <math>f\in \mathcal{PC}</math>, <math>\int_a^b f(x)dx=\tilde{\mathsf{I}}(f)</math>.
 
==The Hahn–Banach theorem==
 
The above theorem can be used to extend a bounded linear transformation <math>T:S\rightarrow Y</math> to a bounded linear transformation from <math>\bar{S}=X</math> to <math>Y</math>, ''if'' <math>S</math> is dense in <math>X</math>. If <math>S</math> is not dense in <math>X</math>, then the [[Hahn–Banach theorem]] may sometimes be used to show that an extension [[existence|exists]]. However, the extension may not be unique.
 
==References==
 
*{{cite book|last=Reed|first=Michael|coauthors=Barry Simon|year=1980|title=Methods of Modern Mathematical Physics, Vol. 1: Functional Analysis|publisher=Academic Press|location=San Diego|isbn=0-12-585050-6}}
 
===Footnotes===
<references/>
 
{{DEFAULTSORT:Continuous Linear Extension}}
[[Category:Functional analysis]]

Latest revision as of 13:32, 25 September 2014

What if you could have a fair and flawless skin that made you look years younger. Eradicating freckles, birth marks and acne marks are other advantages of applying skin whitening cream. If you do not want to eat papaya, you can still whiten your skin using this fruit by using papaya-based soaps, creams, and lotions. The whiter your smile is, the more young and vibrant appearance you will possess; Also, it may get a feel of cosmetic enhancement at a cheaper price. You will probably have to zoom in even further later when working on the finer details, but for now, just pick a 'comfortable' range.

Many people spend a a lot of money on skin whitening lotion and skincare products that claim to remove impurities and lighten skin color. Set it to Lower Field first, and now it's ready for the Topaz filter. Wang and his colleagues have applied for patents in the United States, Japan and Taiwan. Chloasma is generally used to describe skin discolorations caused by hormones. Papaya also contains antioxidants which can promote cell renewal plus could possibly also quit skin aging too.

Eating fresh strawberries can help whiten and brighten your smile, and you can get your tummy full while doing it. Although the contention that parabens in a single product is limited to low levels, an individual uses multiple products as part of personal hygiene and aesthetic enhancements more than once in a single day. Choose this type of antiperspirants and try to opt out of it if you don't really have to, to give your underarms some time to breathe. But most of the time skin needs help so that exfoliation can become a regular event for our skin. Dermatologist examines the skin and focus only skin care regime.

Sugar with honey applied to our skin can exfoliate them just as well as the salons can. Doing this assures that the clay won't sink to the bottom of the soap. All you need is 1 egg, a teaspoon of olive oil, and 1 tissue. The results is an increase of the synthesis of the melanin and the rise of the pigmentation of the skin. We all know that all complexions are wonderful and it matters not which skin color you have, nevertheless for those who have neglected their pale skin color in the past when you had been some sort of sunshine worshipper and also you are somebody who wants some sort of brighter skin color there is nothing wrong within wanting a brighter skin.

Once you have identified the allergy you may have and start to treat it, you may find that your circles will begin to fade. In my experience, baking soda works great as a skin cleanser and mask. Some over the counter face whitening creams can cause damage to your face if misused. Skin bleaching is the practice of applying specific chemical substances or natural recipes to make the skin tone lighter or rather make it even. It is impossible to achieve a healthy white radiance over night.

If you have any queries about exactly where and also tips on how to employ skin brightening cream, you are able to e-mail us in our page.

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