Functional Analysis 19 | Hölder's Inequality

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  • เผยแพร่เมื่อ 24 ธ.ค. 2024

ความคิดเห็น • 20

  • @jaimelima2420
    @jaimelima2420 4 ปีที่แล้ว +16

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

  • @zazinjozaza6193
    @zazinjozaza6193 4 ปีที่แล้ว +9

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

  • @xwyl
    @xwyl 2 ปีที่แล้ว +3

    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
      @wesleyrm ปีที่แล้ว

      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.

  • @BBB32648
    @BBB32648 3 ปีที่แล้ว +1

    Fantastic proof presentation.

  • @homelylad
    @homelylad 3 ปีที่แล้ว +1

    The two proofs were outstanding

  • @tim-701cca
    @tim-701cca ปีที่แล้ว

    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.

  • @zephrias8789
    @zephrias8789 6 หลายเดือนก่อน

    Very good explanation, thank you!

  • @rodrigodiazarancibia5486
    @rodrigodiazarancibia5486 3 หลายเดือนก่อน

    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
      @brightsideofmaths  3 หลายเดือนก่อน +1

      Thanks for the question. Is that even needed here?

    • @rodrigodiazarancibia5486
      @rodrigodiazarancibia5486 3 หลายเดือนก่อน

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

  • @struyep
    @struyep ปีที่แล้ว

    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

  • @darkstudios001
    @darkstudios001 4 ปีที่แล้ว

    Great videos! I love this channel

  • @scienceoftheheart8759
    @scienceoftheheart8759 ปีที่แล้ว

    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?

  • @qiaohuizhou6960
    @qiaohuizhou6960 3 ปีที่แล้ว

    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
      @brightsideofmaths  3 ปีที่แล้ว

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

    • @qiaohuizhou6960
      @qiaohuizhou6960 3 ปีที่แล้ว

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