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)]]

Revision as of 22:49, 2 March 2014

It is a widespread fact that women are obsessed and more aware of their body image as compared to guys. There are furthermore instances whenever a female goes from phases whenever she is not careful regarding what she eats plus does not watch her fat at all. After pregnancy plus whilst the girl is breast feeding her baby, she seems to ignore her health plus there is moreover a tendency towards over eating. What 1 must remember is that both over-obsession and ignorance towards a wellness could spell doom for we.

Tuna is the featured protein on the 3 day Cardiac diet. 1/2 cup of tuna is recommended found on the first 2 days, plus on the final day, you'll grow that to 1 cup. Variety and nutrition is provided with small to medium sized portions of cheese, eggs, crackers, vegetables, grapefruit, banana, and toast. Coffee, tea, or diet soda could accompany food but water ought to be the drink of choice outside of mealtime.

Some folks think it is wise to test plus lose a limited pounds before computing for their body mass index. You should remember which carrying this out sort of thing is similar to cheating during an exam at school. We need to measure your real fat in purchase to receive a well-defined pic of how unhealthy you actually are. If you try to get rid of weight before the computation, we might receive a lower body mass index which will effectively mask your real condition. This might create you think which we don't have to go to the doctor or which we don't have to lose fat at all.

A lot of persons have benefited within the bmi calculator females over the past limited years but, there are more exact alternatives accessible nowadays. If you are not certain what way is ideal for a current condition, then you need to consult a wellness specialist for proper guidance.

Additionally, people vary inside how muscular they are. A very muscular individual will have low body fat plus still have a extremely high BMI. Professional body builders are obese, by this criterion. Arnold Schwarzenegger, inside his prime, was 6'2 and 235 pounds http://www.bodybuildinguniverse.com/arnold.htm; that provides him a BMI of 30.2! There are running backs in American sports whom are 5'8" or 5'9" and over 200 pounds. Running backs are not fat. Heck, Michael Jordan was 6'6" and 216 pounds - borderline fat per the above mentioned!

The 3 Day Cardiac Diet has set menu plans for breakfast, lunch and dinner for three days. Each day is limited to regarding 800 calories. While some claim which there is a specialized "chemical reaction," much of the fat loss benefit is probably obtained by the limited calorie consumption. Additionally, there are truly few carbs on the diet that leads to water reduction.

Yes, of course. BMI is employed to receive a surprisingly general idea of someone's body composition. It should never be employed to diagnose any healthcare condition. If you are a severe athlete or have health issues then it is very suggested which we utilized a more exact way of finding the body fat percentage.

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