Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Equivalence classes

  1. Dec 19, 2014 #1
    \begin{equation*}Let\text{ } (X,\mathcal{A} ,\mu ) \text{ }be \text{ }a \text{ }complete\text{ } \sigma -finite\text{ } measure\text{ } space \\and \text{ }Y \text{ }be \text{ }a \text{ }separable\text{ } Banach\text{ } space\text{ } supplied \text{ }with \text{ }the \text{ }norm\text{ } \left\Vert .\right\Vert . \\For \text{ }every \text{ }p,1\leq p<\infty \text{ } let \text{ } L^{p}(X,Y,\mu ) \text{ }be \text{ }the \text{ }vector \text{ }space \text{ }of \text{ }all \text{ }equivalence \text{ }classes\\ with \text{ }the \text{ }norm \text{ }\left\Vert f\right\Vert _{p}=(\int_{X}\left\Vert f\right\Vert ^{^{^{p}}}d\mu )^{\frac{1}{p}}. \\
    Question: \text{ }How \text{ }do \text{ }I \text{ }writing \text{ }elements\text{ } L^{p}(X,Y,\mu ) \text{ }mathematically?\\
    Thanks
    \end{equation*}
     
  2. jcsd
  3. Dec 21, 2014 #2

    Stephen Tashi

    User Avatar
    Science Advisor

    You need to edit the LaTex so the phrase "of all equivalence classes" shows up completely. As it is, the word "classes" is not visible.

    Is your question: "What is a standard notation for an equivalence class of functions?".

    Or are you asking "What is the definition of an equivalence class of functions with respect to the [itex] ||\ ||_p [/itex] norm?"
     
  4. Dec 22, 2014 #3
    "What is the definition of an equivalence class of functions with respect to the || ||p norm?"
     
  5. Dec 22, 2014 #4

    Stephen Tashi

    User Avatar
    Science Advisor

    My opinion: [itex] f\ =_{p} \ g\iff \ || f - g ||_p = 0 [/itex].
     
  6. Dec 22, 2014 #5
    You can say it in terms of the p-norm, but that's equivalent to the fact that the functions agree almost everywhere. So, a function is equivalent to another function if they agree almost everywhere. This is what you need for the norm to actually be a norm.
     
  7. Dec 22, 2014 #6

    WWGD

    User Avatar
    Science Advisor
    Gold Member

    Like homeo. said, otherwise you will have non-zero vectors with norm zero.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Equivalence classes
  1. Analysis class (Replies: 1)

Loading...