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In [[mathematics]], the '''Freudenthal spectral theorem''' is a result in [[Riesz space|Riesz space theory]] proved by [[Hans Freudenthal]] in 1936. It roughly states that any  element dominated by a positive element in a [[Riesz space]] with the [[Riesz space#Projection properties|principal projection property]] can in a sense be approximated uniformly by [[simple function]]s.


Numerous well-known results may be derived from the Freudenthal spectral theorem. The well-known [[Radon–Nikodym theorem]], the validity of the [[Poisson kernel|Poisson formula]] and the [[spectral theorem]] from the theory of [[normal operator]]s can all be shown to follow as special cases of the Freudenthal spectral theorem.


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Let ''e'' be any positive element in a Riesz space ''E''. A positive element of ''p'' in ''E'' is called a component of ''e'' if <math>p\wedge(e-p)=0</math>. If <math>p_1,p_2,\ldots,p_n</math> are pairwise [[Riesz space#Disjoint complements|disjoint]] components of ''e'', any real linear combination of <math>p_1,p_2,\ldots,p_n</math> is called an ''e''-simple function.
 
The Freudenthal spectral theorem states: Let ''E'' be any Riesz space with the principal projection property and ''e'' any positive element in ''E''. Then for any element ''f'' in the principal ideal generated by ''e'', there exist sequences <math>\{s_n\}</math> and <math>\{t_n\}</math> of ''e''-simple functions, such that  <math>\{s_n\}</math> is monotone increasing and converges [[Riesz space#Order convergence|''e''-uniformly]] to ''f'', and  <math>\{t_n\}</math>  is monotone decreasing and converges ''e''-uniformly to ''f''.
 
== Relation to the Radon–Nikodym theorem ==
Let <math>(X,\Sigma)</math> be a [[measure space]] and <math>M_\sigma</math> the real space of [[signed measure#The space of signed measures|signed <math>\sigma</math>-additive measures]] on <math>(X,\Sigma)</math>. It can be shown that <math>M_\sigma</math> is a [[Dedekind complete]] [[Riesz space|Banach Lattice]] with the [[total variation norm]], and hence has the [[Riesz space#Projection properties|principal projection property]]. For any positive measure <math>\mu</math>, <math>\mu</math>-simple functions (as defined above) can be shown to correspond exactly to <math>\mu</math>-measurable [[simple function]]s on <math>(X,\Sigma)</math> (in the usual sense). Moreover, since by the Freudenthal spectral theorem, any measure <math>\nu</math> in the [[Riesz space#Bands and σ-Ideals|band generated]] by <math>\mu</math> can be monotonously approximated from below by <math>\mu</math>-measurable simple functions on <math>(X,\Sigma)</math>, by [[Monotone convergence theorem#Lebesgue's monotone convergence theorem|Lebesgue's monotone convergence theorem]] <math>\nu</math> can be shown to correspond to an <math>L^1(X,\Sigma,\mu)</math> function and establishes an isometric lattice isomorphism between the band generated by <math>\mu</math> and the Banach Lattice <math>L^1(X,\Sigma,\mu)</math>.
 
==See also==
* [[Radon–Nikodym theorem]]
 
{{Functional Analysis}}
 
==References==
* {{Citation
  | last = Zaanen  | first = Adriaan C.
  | year = 1996
  | title = Introduction to Operator Theory in Riesz spaces
  | publisher = [[Springer Science+Business Media|Springer]]
  | isbn = 3-540-61989-5
}}
* {{Citation
  | last = Zaanen  | first = Adriaan C.
  | last = Luxemburg | first = W. A. J.
  | year = 1971
  | title = Riesz spaces I
  | publisher = [[North-Holland]]
  | isbn = 0-7204-2451-8
}}
 
[[Category:Theorems in functional analysis]]

Latest revision as of 23:00, 6 January 2015


Title:
5 Ways to Make Your BBQ a Memorable Family Event

Word Count:
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Summary:
Every barbecue can and should be an exciting family event. Here are 5 ways to make your routine evening barbecue an experience you'll look back on with a grin.


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Article Body:
Since Americans in the South began roasting pigs publicly, Barbecues have been a staple of North American living. For many, grilling becomes a routine mealtime activity, while for the most devoted grillers, summer is ruled by massive barbecue parties. But grillers do not have to choose between boring repetition and outrageous get-togethers: every barbecue can and should be an exciting family event. Here are 5 ways to make your routine evening barbecue a more unique experience:

1) Who said you have to stick to meat? Much can be done with bread and fruits over an open flame. To make this into a fun activity with your kids, use cookie cutters to make the bread and fruit into shapes. You can decide on themes like "animals" and "letters" or ask that each shape be different from the others! Then, let your kids decide which toppings they would like to put on. Add cheese, onion and various spices to bread and cinnamon and honey to fruit.

2) Try foil-wrapped surprise meals. Place different combinations of chicken, fish and vegetables with sauces and spices in foil packets. These cook in less than 20 minutes and lead to no-mess, lip-smacking results. Children will love the surprise element and can easily switch packets. You can even ask your family for combo suggestions before the barbecue, or draw ingredient names out of a hat to create new meal ideas.

3) A little variety goes a long way. If you tend to stick to the same old beef burgers and pork wieners, why not try grilled pizza or salmon steaks? Similarly, try to mix it up with condiments and seasoning. Sick of ketchup? Try salsa instead. Look online where you'll find hundreds of easy recipes for barbecue rubs. These give your food that extra pizzazz and prevent the mess and drip that comes with barbecue sauces.

4) Barbecues don't have to be rushed. In fact, barbecue devotees will tell you that today's quick broiling is an insult to the term, which should represent a process of slow cooking on low heat. You can find many slow cooker recipes online, and the wait is a great opportunity to pass some quality time with your family and friends. Entertaining backyard activities include creating an outdoor canvas to paint on or playing backyard bowling with household items.

5) Mom said not to play with your food, but there's always an exception! Barbecue games are great fun and tasty at the same time. In the game Glutton, each player gets a certain amount of mashed potatoes, macaroni and cheese or any other side dish. Various kitchen utensils are put in a box and the players blindly pick a few. On the go, they begin to eat their dish with the items that they took out. The first to finish his or her food wins the prize!

Finally, don't fret! Far too many people get lost in the details of planning the perfect barbecue party and forget to enjoy the social atmosphere and delicious food. So kick back, bask in the relaxing sun and prepare http://tinyurl.com/kecvhhb for a summer of unforgettable barbecues!

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