# Proving Density of $S$ in $L^{p'}(E)$ for $g \in L^p(E)$

• MHB
• joypav
In summary: A} |f_n g| = \int_{A} hg = \int_{A} |g|^{p'}. Therefore, by the Lebesgue Dominated Convergence Theorem, we have:\begin{align*}\int_{A} |g|^{p'} = \lim_{n \to \infty} \int_{A} |f_n g| = 1\end{align*}But this contradicts the fact that|g|^{p'}$is a non-zero function on$A$. Therefore, our assumption that$g \neq 0$must be false, and we conclude that$g = 0$in$L^p(E
joypav
Problem:
$E$ is a measurable set and $1 \leq p < \infty$. Let $p′$ be the conjugate of $p$, and $S$ is a dense subset of $L^{p′}(E)$. Show that if $g \in L^p(E)$ and $\int_{E}fg = 0$ for all $f \in S$, then $g= 0$.

Definition of Density:
$S$ is dense in $L^{p'}(E)$ if $\forall h \in L^{p'}(E), \forall \epsilon > 0, \exists f \in S$ s.t. $\left| \left| f-h \right| \right|_{p'} < \epsilon$
or equivalently
$\exists (f_n)$ in $S$ s.t. $\lim_{{n}\to{\infty}}f_n=h$ a.e. on $E$.

Idea?:
As p and p' are conjugates, I was thinking to use Holder's Inequality.
$\int_{E}\left| fg \right| \leq \left| \left| f \right| \right|_{p'} \left| \left| g \right| \right|_p$

Last edited:
Hi joypav,

By density of $S$ in $L^{p'}(E)$, the integral $\int_E fg = 0$ for all $f \in L^{p'}(E)$. Construct $f\in L^{p'}(E)$ such that $fg = \lvert g\rvert^p$ and $\int_E \lvert f\rvert^{p'} \le \int_E \lvert g\rvert^p$. Deduce that $\int_E \lvert g\rvert^p = 0$, i.e., $\|g\|_{L^p(E)} = 0$. Then $g = 0$ in $L^p(E)$.

Yes, using Holder's inequality would be a good idea. Here's how you can use it to prove the statement:

Assume $g \neq 0$. Since $g \in L^p(E)$, there exists a non-zero measurable set $A \subset E$ such that $g|_A \neq 0$. Let $h = \frac{|g|^{p'}}{\left| \left| g \right| \right|_p^{p'-1}}$. Then $h \in L^{p'}(E)$ and $\left| \left| h \right| \right|_{p'} = 1$. Therefore, by the definition of density, there exists a sequence $(f_n)$ in $S$ such that $\lim_{n \to \infty} f_n = h$ a.e. on $E$.

Now, using Holder's inequality, we have:
\begin{align*}
\int_{A} |f_n g| &= \int_{A} |f_n| |g| \\
&\leq \left| \left| f_n \right| \right|_{p'} \left| \left| g \right| \right|_p \\
&= \left| \left| f_n \right| \right|_{p'} \left| \left| g| \right| \right|_p^{p'-1} \left| \left| g \right| \right|_p \\
&= \left| \left| f_n \right| \right|_{p'} \int_{A} h |g| \\
&= \left| \left| f_n \right| \right|_{p'} \int_{A} |g|^{p'} \\
&= \left| \left| f_n \right| \right|_{p'} \left| \left| g \right| \right|_{p'}^{p'} \\
&= \left| \left| f_n \right| \right|_{p'} \\
&\to 1 \text{ as } n \to \infty
\end{align*}

However, since $\lim_{n \to \infty} f_n = h$ a.e. on $E$, we also have $\lim_{n \to \infty} \int ## Related to Proving Density of$S$in$L^{p'}(E)$for$g \in L^p(E)$## 1. What is the definition of density in$L^{p'}(E)$? Density in$L^{p'}(E)$refers to the property of a function or sequence that allows it to be approximated by other functions or sequences in the same space. In other words, a function or sequence is said to be dense in$L^{p'}(E)$if it can be closely approximated by other functions or sequences in the space. ## 2. How is density related to$L^p(E)$? Density in$L^{p'}(E)$is closely related to the space$L^p(E)$, as it is defined as the set of all functions or sequences that are dense in$L^{p'}(E)$. This means that any function or sequence in$L^p(E)$can be approximated by a function or sequence in$L^{p'}(E)$. ## 3. What does it mean to prove density in$L^{p'}(E)$for a function$g \in L^p(E)$? Proving density in$L^{p'}(E)$for a function$g \in L^p(E)$means showing that the function$g$can be closely approximated by other functions in the space$L^{p'}(E)$. This is typically done by constructing a sequence of functions that converges to$g$in the$L^{p'}(E)$norm. ## 4. Why is proving density in$L^{p'}(E)$important? Proving density in$L^{p'}(E)$is important because it allows us to extend the properties of functions in$L^p(E)$to functions in$L^{p'}(E)$. This is useful in many areas of mathematics and physics, where functions in$L^{p'}(E)$are often used as test functions to approximate more complex functions in$L^p(E)$. ## 5. What are some applications of proving density in$L^{p'}(E)$? Proving density in$L^{p'}(E)\$ has many applications in mathematics and physics. For example, it is used to extend the properties of smooth functions to more general functions, to prove the existence and uniqueness of solutions to differential equations, and to study the convergence of Fourier series. It is also an important tool in functional analysis, where it is used to study the properties of various function spaces.

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