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

In summary, the definition of an equivalence class of functions with respect to the \left\Vert .\right\Vert _{p} norm is that two functions are equivalent if they agree almost everywhere. This is necessary for the norm to be a true norm.
  • #1
moh salem
21
0
\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*}
 
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  • #2
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?"
 
  • #3
"What is the definition of an equivalence class of functions with respect to the || ||p norm?"
 
  • #4
My opinion: [itex] f\ =_{p} \ g\iff \ || f - g ||_p = 0 [/itex].
 
  • #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.
 
  • #6
Like homeo. said, otherwise you will have non-zero vectors with norm zero.
 

What is the definition of $L^p(X,Y,\mu)$?

$L^p(X,Y,\mu)$ is a mathematical notation used to describe a function space, where $X$ is the domain, $Y$ is the codomain, and $\mu$ is a measure. The functions in this space are those that are measurable with respect to $\mu$ and have a finite $p$-norm, which is defined as $\|f\|_p = \left(\int_X |f|^p d\mu\right)^{1/p}$ for $p \geq 1$.

What does the $L^p$ norm represent?

The $L^p$ norm, denoted as $\|f\|_p$, is a measure of the size or magnitude of a function in the $L^p$ space. It is defined as the $p$th root of the integral of the function's $p$th power over the domain, with respect to the given measure $\mu$. It is used to compare the sizes of different functions and to study the convergence of sequences of functions in the $L^p$ space.

What is the significance of the measure $\mu$ in $L^p$ spaces?

The measure $\mu$ in $L^p$ spaces plays a crucial role in defining the space and its properties. It is used to determine which functions are measurable and to define the $L^p$ norm. The choice of measure can also impact the behavior of functions in the space, such as their convergence or continuity.

What is the relationship between $L^p$ spaces and Lebesgue spaces?

$L^p$ spaces are a type of Lebesgue space, which is a more general class of function spaces that includes the $L^p$ spaces. Lebesgue spaces are defined using the Lebesgue measure and have additional properties, such as completeness and separability, that may not hold for all $L^p$ spaces.

How are $L^p$ spaces used in mathematical analysis?

$L^p$ spaces are used in mathematical analysis to study the properties of functions and sequences of functions. They are particularly useful in studying the convergence and continuity of functions, as well as in solving differential equations and other mathematical problems. $L^p$ spaces also have applications in other areas of mathematics and in physics, engineering, and other fields.

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