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killThemDonks   Canada. Apr 09 2009 17:05. Posts 2681
kno how to go about proving this?

for any functionals f and g.


//or at least kno of an example where equality doesnt hold? =/

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 Last edit: 09/04/2009 17:08

boblion   Andorra. Apr 09 2009 17:15. Posts 354

f(t) + g(t) =< sup f(t) + sup g(t)

sup ( f(t) + g(t) ) =< sup ( sup f(t) + sup g(t) ) = sup f(t) + sup g(t)


no ?

i'm rusty :<

I worship variance but she is a bitch.Last edit: 09/04/2009 17:15

spencer   United States. Apr 09 2009 17:18. Posts 23

Say S = sup f(t)+g(t), F = sup f(t), G = sup g(t). Note that F > f(t), G > g(t) for all t in A so that F+G is an upper bound for f+g, but if S > F+G, the property of the supremum implies F+G is not an upper bound for f+g.


spencer   United States. Apr 09 2009 17:22. Posts 23

the key idea is that supremum is the least upper bound, so once you find an upper bound, you know the supremum has to be <= to it. thats basically what boblion did too.


killThemDonks   Canada. Apr 09 2009 17:24. Posts 2681

heh..that was easy -_-...ya im rusty too. any examples of functions where equality doesnt hold? equality should hold for all continuous functions i think


spencer   United States. Apr 09 2009 17:29. Posts 23

im pretty sure it's true for all real functions. the proof assumes no special properties about f or g.

edit: misread your question, so ignore this

 Last edit: 09/04/2009 19:00

k3rn3l   Hungary. Apr 09 2009 17:31. Posts 217

 Last edit: 09/04/2009 17:31

boblion   Andorra. Apr 09 2009 17:31. Posts 354


  On April 09 2009 16:24 killThemDonks wrote:
heh..that was easy -_-...ya im rusty too. any examples of functions where equality doesnt hold?


No idea but i guess it is always true.


  On April 09 2009 16:24 killThemDonks wrote:
equality should hold for all continuous functions i think


For sure.

I worship variance but she is a bitch. 

killThemDonks   Canada. Apr 09 2009 17:35. Posts 2681

are any of u in pure math? i may have some tougher questions..not homework or anything...just a refresher


boblion   Andorra. Apr 09 2009 17:38. Posts 354


  On April 09 2009 16:35 killThemDonks wrote:
are any of u in pure math? i may have some tougher questions..not homework or anything...just a refresher


I'm math/eco but i'm bad and rusty

I worship variance but she is a bitch. 

spencer   United States. Apr 09 2009 17:41. Posts 23

yeah I am, just finished a class in real analysis last semester actually


killThemDonks   Canada. Apr 09 2009 17:58. Posts 2681

hmm wait....equality holds for continuous functions in general....or continuous functions over a closed set...?? bah...soo rusty


Zalfor   United States. Apr 09 2009 18:03. Posts 2236

i'm a math major.

equality holds


killThemDonks   Canada. Apr 09 2009 18:04. Posts 2681

no wait...im an idiot...brb


killThemDonks   Canada. Apr 09 2009 18:14. Posts 2681

now im confused...cuz say f, and g are continuous over some (closed) set T,
=> sup{ f(t): t in T} = max{ f(t): t in T} ...same holds for g(t)

Let h(t) = f(t) + g(t), it follows that h(t) is also continuous over T ==> sup h(t) = max h(t) = max( f(t) + g(t) ) =/= max( f(t) ) + max ( g(t) )

:S.. amirite?


take functions N(0,1) and N(1,1) for example over T = [-2,2]

so equality doesnt hold for cts functions????


math is idiotic.....


lazymej   Canada. Apr 09 2009 18:21. Posts 2897

lol math


killThemDonks   Canada. Apr 09 2009 18:34. Posts 2681


  On April 09 2009 17:14 killThemDonks wrote:
now im confused...cuz say f, and g are continuous over some (closed) set T,
=> sup{ f(t): t in T} = max{ f(t): t in T} ...same holds for g(t)

Let h(t) = f(t) + g(t), it follows that h(t) is also continuous over T ==> sup h(t) = max h(t) = max( f(t) + g(t) ) =/= max( f(t) ) + max ( g(t) )

:S.. amirite?


take functions N(0,1) and N(1,1) for example over T = [-2,2]

so equality doesnt hold for cts functions????


math is idiotic.....




halp!


spencer   United States. Apr 09 2009 18:52. Posts 23

yeah, equality definitely doesn't hold even for continuous functions in general - it's true only if the maxes of f and g coincide at the same point


Day[9]   United States. Apr 09 2009 18:56. Posts 3447

here's an easy one

f(t) = cos t
g(t) = -cos t

individually, each one oscillates between 1 and -1, thus each has a supremum of 1

add them together, and you get f(t) + g(t) = 0. the supremum is 0

zing


Day[9]   United States. Apr 09 2009 18:57. Posts 3447

that is equality doesn't hold for those bad boys


Day[9]   United States. Apr 09 2009 18:58. Posts 3447

that is sup ( f(t) + g(t) ) = sup ( 0 ) = 0 < sup f(t) + sup g(t) = 1+1 = 2

0 < 2

eh?


killThemDonks   Canada. Apr 09 2009 19:06. Posts 2681

yups!


NiTE   Croatia. Apr 09 2009 19:12. Posts 366


  On April 09 2009 16:31 k3rn3l wrote:


I lol'd

Have an opinion about what I said? please say it 

boblion   Andorra. Apr 09 2009 19:42. Posts 354


  On April 09 2009 17:58 Day[9] wrote:
that is sup ( f(t) + g(t) ) = sup ( 0 ) = 0 < sup f(t) + sup g(t) = 1+1 = 2

0 < 2

eh?



:
So what ?
o,o
the equality holds o,o

Or i'm really tired, stupid and Romanian.

Day u Fail

I worship variance but she is a bitch.Last edit: 09/04/2009 19:46

killThemDonks   Canada. Apr 09 2009 20:05. Posts 2681

^ equality doesnt hold necessarily for continuous functions (not sure why i thought so in the first place :S).


boblion   Andorra. Apr 09 2009 20:17. Posts 354


i need sleep maybe ?

I worship variance but she is a bitch.Last edit: 09/04/2009 20:18

boblion   Andorra. Apr 09 2009 20:23. Posts 354

omg

i'm retarded. I thought Day said that the whole "=<" was wrong lol

I worship variance but she is a bitch. 

Bigbobm   United States. Apr 15 2009 04:38. Posts 5512

day9 is smart
and right

Its time to stop thinking like a bitch and think smart like a poker player - ket 

rockman255   Canada. Apr 16 2009 01:07. Posts 4471

how did i receive a degree in mathematics O___o

rockman255: its not easy being superman U N0 MySteeZ: mega man. rockman255: same thing U N0 MySteeZ: no 

xx frankie   United States. Apr 16 2009 11:34. Posts 57

E=Mc2


PokerDoc88   Australia. Apr 16 2009 13:37. Posts 3527


  On April 09 2009 18:12 NiTE wrote:
Show nested quote +


I lol'd


LMFAO this is the first thing I thought when I saw the equation. Well, not exactly a cat saying "sup" but definitely the "Sup?" mentality came to mind.

Wish I could help, I haven't touched any math for 3 years and this looks beyond anything I knew even when I was doing this back in high school. I miss math...


 



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