Hölders inequality proof
Hi I've come across the Hölders inequality and I was wondering if there was a proof that it's true that is relatively simple (teacher can't remember the proof).
I understand the Cauchy-Schwarz inequality if it helps.
I've also attached a photo of the maths book it's in. You can ignore from the now suppose .... As that's the explanation of how you reach the rearrangement and Chebyshev's inequality .
Google isn't coming up with anything other that Young's inequality and we haven't done that.
I understand the Cauchy-Schwarz inequality if it helps.
I've also attached a photo of the maths book it's in. You can ignore from the now suppose .... As that's the explanation of how you reach the rearrangement and Chebyshev's inequality .
Google isn't coming up with anything other that Young's inequality and we haven't done that.
(edited 3 months ago)
Original post by Omnibluey
This is my attempt at mucking about with it
The hope was the x1y1 and x2y2 terms would lead to an equality and the remaining terms would lead to the greater than. It's not working
This is my attempt at mucking about with it
The hope was the x1y1 and x2y2 terms would lead to an equality and the remaining terms would lead to the greater than. It's not working
As in I'm aware the photo isn't attaching properly
It helps to see what you did/your book said (realise you tried), but a relatively simple walkthrough of Holder is
https://www.youtube.com/watch?v=kxQiKaIuyOg&ab_channel=DrChrisTisdell
and while it does use Youngs, thats not that hard to establish first.
https://www.youtube.com/watch?v=kxQiKaIuyOg&ab_channel=DrChrisTisdell
and while it does use Youngs, thats not that hard to establish first.
The book explains Cauchy-Schwarz as being proved by a dot product but where the cosine is always 1 on the greater than side. (I can follow that part perfectly fine). And then just went hölders is a generalisation. I think the book picture attached fine.
(It's in the UKMT Olympiad companion book Pg 12 if you have it)
(It's in the UKMT Olympiad companion book Pg 12 if you have it)
(edited 3 months ago)
Original post by Omnibluey
The book explains Cauchy-Schwarz as being proved by a dot product but where the cosine is always 1 on the greater than side. (I can follow that part perfectly fine). And then just went hölders is a generalisation. I think the book picture attached fine.
(It's in the UKMT Olympiad companion book Pg 12 if you have it)
(It's in the UKMT Olympiad companion book Pg 12 if you have it)
The youtube explanation divides through by the norm of the vectors so they become unit vectors and then youre proving its <= 1 which is effectively the cos bound. He then shows that using calculus/youngs which isnt too bad to follow but there are other (similar) ways of doing that part. If its ukmt theyre not into calculus, you can take a convex argument
https://people.bath.ac.uk/masmdp/measdir.bho/notesp54to61.pdf
or ...
(edited 3 months ago)
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