Functional Analysis 19 | Hölder's Inequality

[@zazinjozaza6193]

So cool that you are making so many videos on functional analysis.

[@jaimelima2420]

Thanks again for putting this together in a clear way. Perhaps two ideas for future series could be "Convex Sets" and "Convex Analysis" ...

[@brightsideofmaths]

On my To-Do-List :)

[@BBB32648]

Fantastic proof presentation.

[@tim-701cca]

I saw an exercise to prove young’s inequality in a book. Consider the graph y=x^{p-1} and the line x=a,y=b.

[@homelylad]

The two proofs were outstanding

[@xwyl]

The proof of Young's inequality is clean and swift, though by construction. Construction is perhaps the way of mathematicians to give a clean proof, but it's not for leaners to deepen their understanding. However, Young's inequality isn't the new knowledge here, and the construction uses only very common concepts. This construction is of the bright side.

[@wesleyrm]

By construction also saves time! If you remember the starting outline of a proof by construction, you can prove it yourself again in the future, better than memorizing the results themselves.

[@struyep]

Young's inequality also follows from the weighted AM-GM inequality. Let x = a^p, y = b^q, u = 1/p, v = 1/q; we have u, v > 0 and u + v = 1. Apply AM-GM with u and v as weights for the elements x and y. Then x^u y^v

[@rodrigodiazarancibia5486]

Thanks so much for the video. I just have one question. In the proof of Young's inequality. How can we justify that lambda reaches the values ​​0 and 1?

[@brightsideofmaths]

Thanks for the question. Is that even needed here?

[@rodrigodiazarancibia5486]

@@brightsideofmaths Ouhh thanks thanks, It is not needed. Now I see it 👌.

[@zephrias8789]

Very good explanation, thank you!

[@brightsideofmaths]

Glad you enjoyed it!

[@scienceoftheheart8759]

Thanks a lot it helps a lot, you said you use obs to record the videos but what about to writes the math exercises? Any drawing program?

[@brightsideofmaths]

LaTeX :)

[@darkstudios001]

Great videos! I love this channel

[@qiaohuizhou6960]

Hi, thank you so much for your video! I am wondering if you could add a video on proving this Holder's inequality on functions defined on measure space? I really have problems understanding what exactly does it mean to have functions defined on an abstract measure space. Is that the measurable function mapping X from abstract measure space to real-valued space, or does it mean a function like L(μ)? I am very puzzled why the x, y seems are just variable defined on R can be substituted by |X| |Y| and the inequality still holds. I hope my question makes sense! thank you so much! Also, in our lecture note, the holder's inequality is proved using convexity inequality, are convexity inequality and Yong's inequality somehow connected?

[@brightsideofmaths]

The ideas stay the same but you are completely right: I should do a video about these abstract concepts!

[@qiaohuizhou6960]

@@brightsideofmaths Thank you so much! That would be super helpful !