If $f$ is a function such that $f \in L^\infty \cap L^ {p_0}$ where $L^\infty$ is the space of essentially bounded functions and $ 0 &lt; p_0 &lt; \infty$. Show that $ || f|| _{L^p} \to ||f || _{L^\infty} $ as $ p \to \infty$. Where $|| f||_{L^\infty} $ is the least $M \in R$ such that $|f(x)| \le M$ for almost every $x \in X$.  The hint says to use the monotone convergence theorem, but i can't even see any pointwise convergence of functions. Any help is appreciated.
5458
Could someone help me prove that given a finite measure space $(X, \mathcal{M}, \sigma)$ and a measurable function $f:X\to\mathbb{R}$ in $L^\infty$ and some $L^q$, $\displaystyle\lim_{p\to\infty}\|f\|_p=\|f\|_\infty$?    I don't know where to start.
42306
This is a problem from Rudin's Functional Analysis:  Show that there is a sequence of functions $\{ f_n \}$ in $X$ (the vector space of all complex functions) such that $f_n \to 0$ pointwise, but if any sequence $\{ \gamma_n \}$ tends to $\infty$, $\gamma_n f_n \not\to 0$.  The hint says to use the fact that the collections of all complex sequences converging to $0$ has the same cardinality as $[0,1]$. Honestly, this confused me more than it helped me.  My main issue is through the method of showing existence: I am not sure if I should attempt to construct an explicit sequence $f_n$, or if I need to appeal to other nonconstructive means such as Zorn's Lemma, etc. I have been trying to construct clever choices of sequences for a good bit now, but it always seems like I will be able to find a sequence that diverges slowly enough to allow that $\gamma_n f_n \to 0$.   Thanks!
111182
I have to solve this problem.      Let $\{\lambda_n\}$ be a sequence of real number such that $\lim_{n\rightarrow\infty}\lambda_n=0$ and consider the operator $T:l^p\rightarrow l^p$, $1\leq p\leq \infty$, defined by      $$T(\{x_1,\ldots x_n,\ldots\})=\{\lambda_1 x_1,\ldots,\lambda_n x_n,\ldots\} $$   Prove that $T$ is compact.    My attempt is the following.  I have considered the operator $$ T_N:l^p\rightarrow l^p,\quad T_N(\{x_1,\ldots x_n,\ldots\})=\{\lambda_1 x_1,\ldots,\lambda_N x_N,0,0,\ldots\}. $$ $T_N$ is clearly linear and it is also compact. In fact given any bounded sequence $\{x^{(n)}\}_{n\geq 1}$ in $l^p$, the sequence $\{T_Nx^{(n)}\}_{n\geq 1}$ is bounded in $\mathbb{R}^N\subset l^p$. Then, since every bounded sequence in $\mathbb{R}^N$ has a convergent subsequence, it follows that $T_N$ is compact.   Now, I have to prove that $T$ is compact. For this I proved that $||T_N-T||\rightarrow 0$ as $N\rightarrow\infty$. In fact we have: $$ ||T_N-T||=\sup_{||x||=1}||(T_N-T)x||=\sup_{||x||=1}||T_Nx-Tx||=\sup_{||x||=1}\left(\sum_{i=N+1}^{\infty}|\lambda_i x_i|^p\right)^{\frac{1}{p}} $$ and, given $\epsilon&gt;0$, there exists $\bar n$ such that $|\lambda_n|&lt;\epsilon$, for every $n&gt;\bar n$. So if we take $N&gt;\bar n$, we get $$ \sup_{||x||=1}\left(\sum_{i=N+1}^{\infty}|\lambda_i x_i|^p\right)^{\frac{1}{p}}\leq\epsilon\sup_{||x||=1}\left(\sum_{i=N+1}^{\infty}|x_i|^p\right)^{\frac{1}{p}}\leq\epsilon $$ Therefore $||T_N-T||\rightarrow 0$ as $N\rightarrow\infty$ implies that $T$ is compact.   My questions are:  (1) is the proof of the compactness of the operator $T_N$ correct?  (2) is the conclusion correct?  Thanks
43867
I ran into the following problem when I was doing my homework, and I have no thoughts on where I should start with:  (1) If $f\in L^{2}$, show that $\displaystyle \lim_{p \rightarrow 1^{+}}\int_{[0,1]}|f|^{p}=\int_{[0,1]}|f|$  (2) If $0&lt;p$, show that $\displaystyle \lim_{q\rightarrow p^{-}}||f||_{q}=||f||_{p}$  My first thought was Generalized LDCT, but it didn't seem to work. I also made some other attempts but none of them were successful... Can anybody give me some hints on how I should look at this question?  Also, I know if $p\rightarrow\infty$ then $||f||_{p}\rightarrow||f||_{\infty}$ on $[0,1]$, but does similar continuity in p holds for other $L^{p}[0,1]$ norms in general?  Thank you!  Edit:   Sorry if I did not make it clear enough in the question. All $L^{p}$ refers to $L^p[0,1]$.  The first question is found  (thanks to t.b.), but the second question remains, mainly because $f$ is not guaranteed to be in any $L^{p}$.
24292
I'm wondering if it's possible for a function to be an $L^p$ space for only one value of $p \in [1,\infty)$ (on either a bounded domain or an unbounded domain).  One can use interpolation to show that if a function is in two $L^p$ spaces, (e.g. $p_1$ and $p_2$,with $p_1 \leq p_2$ then it is in all $p_1\leq p \leq p_2$).   Moreover, if we're on a bounded domain, we also have the relatively standard result that if $f \in L^{p_1}$ for some $p_1 \in [1,\infty)$, then it is in $L^p$ for every $p\leq p_1$ (which can be shown using H&ouml;lder's inequality).  Thus, I think that the question can be reduced to unbounded domains if we consider the question for any $p&gt;1$.   Intuitively, a function on an unbounded domain is inside an $L^p$ space if it decrease quickly enough toward infinity. This makes it seem like we might be able to multiply the function by a slightly larger exponent. At the same time, doing this might cause the function to blow up near zero. That's not precise/rigorous at all though.     So I'm wondering if it is possible to either construct an example or prove that this can't be true.
51435
I'm having trouble with this problem from an old analysis qual:  Find a function $f$ such that for $p\in (1,\infty)$, $f$ is in $L^p(\mathbb{R})$ only when $p=4$.
98512
I want to show that if $f$ is non-increasing and $f\in L_{1}([a,\infty),m)$ where $m$ is Lebesgue measure then $\lim_{t\to\infty} t f(t)=0$. So far I've been able to show that $f\geq 0$ and that $\lim_{t\to\infty} f(t)=0$. Since monotone functions are differentiable a.e. I thought about using integration by parts but couldn't get anywhere with that. Any hints or suggestions would be greatly appreciated.
24565
Let $f:\mathbb{R}\to\mathbb{R}$ continuous such that $\lim_{x\to\infty}f(x+1)-f(x)=l$.  How can I prove that $\dfrac{f(x)}x$ converges for $x\to+\infty$ ?  I am just trying to prove the convergence, not that it converges to $l$ (that part be deduced immediately by Stolz-Cesaro and the sequential characterization of limits)
68882
The question is: if $f_i$ is a sequence of functions in $L^p$ converging to $f$ and $g_i$ a sequence in $L^q$ converging to $g$ show that $f_ig_i$ converges to $fg$ in $L^1$ for $p,q$ finite and $\frac{1}{p}+\frac{1}{q}=1$. Does this result hold if $p=1, q=\infty$?  So I think I showed the first part. $||f_ig_i-fg||_1=||f_ig_i-f_ig+f_ig-fg||_1\leq||f_ig_i-f_ig||_1+||f_ig-fg||_1 = ||f_i(g_i-g)||_1+||g(f_i-f)||_1\leq||f_i||_p||g_i-g||_q+||g||_q||f_i-f||_p$   (by Holder's inequality) which goes to $0$ as $i\rightarrow\infty$.  However the second part of the question throws me off because I don't see why this proof doesn't work just as well for $p=1,q=\infty$.  Am I missing something? Does the second part also hold?
83152
Is there some simple characterization of weak convergence of sequences in the space $\ell_\infty$? If yes, is there some similar claim for nets?   I was only able to come up with a characterization of sequential weak convergence using limit along ultrafilters, which I will describe below. I wonder whether there is some insight into this characterization. (E.g. whether there is some simple reformulations which does not use ultralimits.) Moreover, I do not know whether at least this characterization works for nets, too.  We have the following result describing weak convergence in $C(K)$, see e.g. Corollary 3.138,  in Banach Space Theory by Fabian, Habala et al. (It is a consequence of Rainwater theorem and characterization of extreme points of unit ball in $C(K)^*$.)     Let $K$ be a compact topological space. Let $\{f_n\}$ be a bounded sequence in $C(K)$ and $f\in C(K)$. Then, if $f_n\to f$ pointwise, we have $f_n\overset{w}\to f$.   Moreover we have isometric isomorphism between $\ell_\infty$ and $C(\beta\mathbb N)$, which is described e.g. in the Wikipedia article on  or in  of Carothers' book A short course on Banach space theory. This isomorphism assigns to each bounded sequence $(x_n)$ the continuous function $\overline x$ on $\beta\mathbb N$ defined by $$\overline x(\mathscr U) = \operatorname{\mathscr U-lim} x_n,$$ where $\operatorname{\mathscr U-lim} x_n$ denotes the  of $x_n$ w.r.t the ultrafilter $\mathscr U$.  Combining the above results we get the following characterization:      Let $f^{(n)},f\in\ell_\infty$. The sequence $f^{(n)}$ converges to $f$ weakly if and only if for every ultrafilter $\mathscr U$    $$\lim_{n\to\infty} \operatorname{\mathscr U-lim} f^{(n)}= \operatorname{\mathscr U-lim} f.$$   (The above claim for principal ultrafilters is just a pointwise convergence. But in the above claim the equality is required for free ultrafilters, too.)
97797
Let $(X, M, m)$ be an arbitrary measure space. Let $f_n, f \in L^1_m(X)$. Assume that $$\int_X |f_n - f| \, dm \leq \frac{1}{n^2} \text{  for all }n \geq 1. $$ Then I want to show that $f_n \rightarrow f$ a.e. on $X$.    I thought since we have norm convergence, we get a subsequence $f_{n_k}$ of $f_n$ converging to $f$ a.e. Does the condition $\int_X |f_n - f| \,dm \leq \frac{1}{n^2}$ for all $n \geq 1$ imply something stronger than norm convergence?
73546
It is stated in my professor's notes that, given a sequence $\{f_j\}$ of  $C_0^\infty(\Omega)$ functions (infinitely differentiable with compact support), and a function $g\in C_0^\infty(\Omega)$, all defined in an open set $\Omega\in\mathbb{R}^n$:  $$\|f_j-g\|_{L_p(\Omega)}\to0\implies|f_j(x)-g(x)|\to0\,\forall x\in\Omega$$  I was not able to prove it, however. Searching on the Internet, I found many counterexamples to the statement that $L^p$-convergence implies pointwise convergence, but they do not deal with $C_0^\infty(\Omega)$ functions, so it may also be true. Can anyone point to a proof or a counterexample?
114628
From Donald Cohn's Measure Theory, section 2.4, exercise 10.  Let $(X, A, \mu)$ be a measure space, and let $f$ and $f_1, f_2, \dots$ be non-negative functions that belong to $L^1(X, A, \mu, R)$ and satisfy (i) $\{f_n\}_n$ converges to $f$ almost everywhere; (ii) $\int fd\mu = \lim_n\int f_nd\mu$. Show that $\lim_n\int |f_n - f|d\mu = 0$.  I let $f_n = 1/n$ on $[n, n+1]$ and $0$ elsewhere, and let $f=0$, then all the convergence theorems in that section (dominated convergence, and monotone convergence) failed.
100455
The following is an real analysis qualifying exam problem that I cannot solve:  Suppose $X$ is a Banach space and that $(x_n)$ converges weakly to $x$. Show that $\liminf ||x_n|| \geq ||x||$.  Using the Uniform Boundedness Principle I can show that $\sup_{n \in \mathbb{N}} ||x_n -x||$ is finite. Using Alaoglu's Theorem I can show that some subset of $(x_n)$ converges in norm to $x$. I feel like I am close with this but cannot seem to finish the problem.
1759
Suppose that $\|f\|_p &lt \infty$ for all $1\leq p &lt p'$, I want to know if the the following is true and in that case how to show it     $p \mapsto \|f\|_p$ is continuous on $[1,p')$   Or maybe we need to impose some more constraints such as finite measure space. In case of finite measure space, I tried to use .
74329
Stuck with this problem from Zgymund's book.   Suppose that $f_{n} \rightarrow f$ almost everywhere and that $f_{n}, f \in L^{p}$ where $1&lt;p&lt;\infty$. Assume that $\|f_{n}\|_{p} \leq M &lt; \infty$. Prove that:  $\int f_{n}g \rightarrow \int fg$ as $n \rightarrow \infty$ for all $g \in L^{q}$ such that $\dfrac{1}{p} + \dfrac{1}{q} = 1$.  Right, so I estimate the difference of the integrals and using Hölder end up with:  $$\left|\int f_{n} g - \int fg\right| \leq \|g\|_{q} \|f_{n} - f\|_{p}$$  From here I'm stuck because we are not assuming convergence in the seminorm but just pointwise convergence almost everywhere. How to proceed?
22963
If we are in a sequence space, then the $ l^{p} $-norm of the sequence $ \mathbf{x} = (x_{i})_{i \in \mathbb{N}} $ is $ \displaystyle \left( \sum_{i=1}^{\infty} |x_{i}|^{p} \right)^{1/p} $.  The $ l^{\infty} $-norm of $ \mathbf{x} $ is $ \displaystyle \sup_{i \in \mathbb{N}} |x_{i}| $.  Prove that the limit of the $ l^{p} $-norms is the $ l^{\infty} $-norm.  I saw an answer for $ L^{p} $-spaces, but I need one for $ l^{p} $-spaces. Besides, I didn’t really understand the $ L^{p} $-answer either.  Thanks for your help!