Quadratic residue: Difference between revisions

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{{for|F-spaces in general topology|sub-Stonean space}}
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In [[functional analysis]], an '''F-space''' is a [[vector space]] ''V'' over the [[real number|real]] or [[complex number|complex]] numbers together with a [[metric (mathematics)|metric]] ''d'' : ''V'' × ''V'' → '''R''' so that
# Scalar multiplication in ''V'' is [[continuous function#Continuous functions between metric spaces|continuous]] with respect to ''d'' and the standard metric on '''R''' or '''C'''.
# Addition in ''V'' is continuous  with respect to ''d''.
# The metric is [[translation-invariant metric|translation-invariant]]; i.e., ''d''(''x'' + ''a'', ''y'' + ''a'') = ''d''(''x'', ''y'') for all ''x'', ''y'' and ''a'' in ''V''
# The metric space (''V'', ''d'') is [[completeness (topology)|complete]]
 
Some authors call these spaces ''[[Fréchet space]]s'', but usually the term is reserved for [[locally convex]] F-spaces. The metric may or may not necessarily be part of the structure on an F-space; many authors only require that such a space be [[metrizable]] in a manner that satisfies the above properties.
 
== Examples ==
 
Clearly, all [[Banach space]]s and [[Fréchet space]]s are F-spaces. In particular, a Banach space is an F-space with an additional requirement that {{nowrap|''d''(''αx'', 0) {{=}} {{!}}α{{!}}⋅''d''(''x'', 0)}}.<ref>Dunford N., Schwartz J.T. (1958). Linear operators. Part I: general theory. Interscience publishers, inc., New York. p. 59</ref>
 
The [[Lp space|L<sup>p</sup> spaces]] are F-spaces for all {{nowrap|p > 0}} and for {{nowrap|p ≥ 1}} they are locally convex and thus Fréchet spaces and even Banach spaces.
 
=== Example 1 ===
<math>\scriptstyle L^\frac{1}{2}[0,\, 1]</math> is a F-space. It admits no continuous seminorms and no continuous linear functionals — it has trivial [[dual space]].
 
=== Example 2 ===
Let <math>\scriptstyle W_p(\mathbb{D})</math> be the space of all complex valued [[Taylor series]]
 
:<math>f(z)=\sum_{n \geq 0}a_n z^n</math>
 
on the unit disc <math>\scriptstyle \mathbb{D}</math> such that
 
:<math>\sum_{n}|a_n|^p < \infty</math>
 
then (for {{nowrap|0 < p < 1}}) <math>\scriptstyle W_p(\mathbb{D})</math> are F-spaces under the [[p-norm]]:
 
:<math>\|f\|_p= \sum_{n}|a_n|^p \qquad (0 < p < 1)</math>
 
In fact, <math>\scriptstyle W_p</math> is a [[quasi-Banach algebra]]. Moreover, for any <math>\scriptstyle \zeta</math> with <math>\scriptstyle |\zeta| \;\leq\; 1</math> the map <math>\scriptstyle f \,\mapsto\, f(\zeta)</math> is a bounded linear (multiplicative functional) on <math>\scriptstyle W_p(\mathbb{D})</math>.
 
== References ==
{{reflist}}
 
* {{citation|first=Walter|last=Rudin|authorlink=Walter Rudin|title=Real & Complex Analysis|publisher=McGraw-Hill|year=1966|isbn=0-07-054234-1}}
 
{{Functional Analysis}}
 
[[Category:Functional analysis]]
[[Category:Topological vector spaces]]
[[Category:F-spaces]]
 
[[pl:Przestrzeń Frécheta (analiza funkcjonalna)]]

Latest revision as of 23:15, 3 October 2014

The thought of consuming green coffee bean extract to begin losing weight might possibly appear truly new yet it has absolutely assisted a amazing deal of individuals return to their desirable weight plus more. Losing fat is never a simple thing to strive for. Besides the undeniable fact that it usually need a lot of effort and discipline, one is not truly certain when the countless products being presented available is effective o-r not. The question today is, can this extract be an perfect method to lose excess weight? Can it be secure? How swiftly can one really see results? Keep reading plus discover.

The initially system of action is the green coffee weight loss ingredient: chlorogenic acid. Chlorogenic acid is a natural phytochemical. Chlorogenic acid works by inhibiting the absorption and the utilization of glucose, moreover known as glucose, into the blood stream. No sugar inside the bloodstream signifies no build up of fat inside the body because glucose turns to fat inside the body when it really is not burned.

When roasted, the beans have their peak flavor from 24 hours to 7 days. The green dried coffee beans could be stored for a longer period before roasting and hold the taste of the bean. Home coffee roasting has the benefit of roasting tiny quantities of green coffee at a time. Storing them for a short period plus then grinding plus brewing as needed.

So far, all of the sounds a bit technical, plus we absolutely only wish to understand when this stuff is any wise. The answer is "Yes", this might be the real thing, plus I might tell you why. Many clinical studies have shown which Chlorogenic Acid slows fat absorption from food expenditure, plus also improves fat metabolism. Don't look to brewed coffee as a source of Chlorogenic Acid, it really is roasted right from the bean.

Not startlingly, I'm good with this aid. Listen, I smell a rat. I'm not a wizard in which. As others have stated, this really is where a few beginning fat reduction fans make their just mistake. It pertaining to this. Definitely, you read it right.

There is not any miracle plus no magic. These are simply unroasted coffee means that have their wellness advantages intact. Usually whenever coffee beans are roasted, Chlorogenic Acid inside the beans is removed.

It has more caffeine than compromise mostly darker counterpart. Axis a cup of coffee beans roasted more caffeine burned off a lighter roast to compromise.

[4] Lafay S., et al, Absorption and metabolism of caffeic acid plus chlorogenic acid inside the small intestine of rats. British Journal of Nutrition, 2006. 96: p39-46.

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