Show that $g_n(x)=ne^{-nx}$ is bounded, where $x\in\Omega=]0,1[$ for $n\in\Bbb{N}.$

Question

Show that $g_n(x)=ne^{-nx}$ is bounded, where $x\in\Omega=]0,1[$ for $n\in\Bbb{N}.$

My trial

I'm thought of it this way that if $g_n(x)=ne^{-nx}$ converges weakly in $L_1(\Omega)$ then it is bounded.

So, I claimed that $g_n\in L_1(\Omega),$ converges to $0$ for each $n\in\Bbb{N}$. This has been confirmed untrue but the following is my previous work.

Now, for each $n,$ \begin{align} \int_{\Omega}|g_n|&=\int_{\Omega}|ne^{-nx}|dx\\&=\int_{]0,1[}ne^{-nx}dx\\&=\int_{[0,1]}ne^{-nx}dx\\&=\int^1_{0}ne^{-nx}dx\\&=1-e^{-n}<\infty\end{align} Thus, $g_n\in L_1(\Omega),$ for each $n.$ Then, $g_n(x)=ne^{-nx}$ converges weakly to $0$ if and only if $F(g_n(x))\to F(0)=0,\;\;\forall\,F\in (L_1(\Omega))^*=L_\infty(\Omega).$ The spaces $(L_1(\Omega))^*$ and $L_\infty(\Omega)$ are equal, since there exists a map that is isometrically isomorphic between $L_\infty(\Omega)$ and $(L_1(\Omega))^*$.

Let $F\in (L_1(\Omega))^*=L_\infty(\Omega).$ However, $L_\infty(\Omega)$ does not have a countable dense subset unlike $L_1(\Omega)$, such that $L_1(\Omega)=\overline{D(\Omega)}.$ So, I don't know how to approximate $F.$

I'm stuck here. Anyone who knows the way out or who can help? I'll appreciate it! Thanks!

Answer 1

First of all the statement $(g_n)$ converges weakly to $0$ makes no sense. Apparently, you are considering $g_n$'s as elements of $L^{1}([0,1])$ and you are talking about weak convergence in this space.

Weak convergence to $0$ in $L^{1}([0,1])$ means $\int_0^{1} g_n(x) h(x)\, dx \to 0$ as $ n \to \infty$ for every $h \in L^{\infty}([0,1])$. Taking $h(x)=1$ you can easily see that this is not true.

Answer 2

For any given $n$, the function $g_n(x)$ is bounded as a function of $x$ on $[0,1]$.

For any fixed $x\in (0,1]$, the sequence $f(n)=g_n(x)$ is bounded as a function of $n$ - because it converges to zero.

However, considered as functions, the sequence $g_n$ fail to converge uniformly or in $L^1[0,1]$; their weak limit is the Dirac delta.

So then, go back to your source and make sure exactly what the question is asking. I suspect it's the question of my second line - the sequence of values at some fixed $x$.