Writing Elements in $L^p(X,Y,\mu)$ Mathematically

[moh salem]

\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*}

 

[Stephen Tashi]

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 $||\ ||_p$ norm?"

 

[moh salem]

"What is the definition of an equivalence class of functions with respect to the || ||p norm?"

 

[Stephen Tashi]

My opinion: $f\ =_{p} \ g\iff \ || f - g ||_p = 0$.

 

[homeomorphic]

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.

 

[WWGD]

Like homeo. said, otherwise you will have non-zero vectors with norm zero.