# Tagged Questions

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### For $1 \leq r < p < \infty$ prove the continuous injection of $L^p([0, 1])$ into $L^r([0, 1])$.

For $1 \leq r < p < \infty$ prove the continuous injection of $L^p([0, 1])$ into $L^r([0, 1])$. I am having a hard time starting. Any suggestions. I tried a straight forward approach. That ...
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### $L^\infty(\Omega)$ space

Consider Lebesgue spaces $L^p(\Omega)$, $\Omega$ is a bounded domain. Let $f \in L^p(\Omega)$ for all $p$. Is it true that $f \in L^\infty(\Omega)$?
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### What does it mean to be an L^1 function?

I am struggling to understand what the space L^1 is, and what it means for a function to be L^1. A friend told me that a function f is $L^1$ if $\int_\mathbb{R} |f|$ is finite. It is $L^2$ if ...
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### On $C_c^{\infty}$ being dense in $L^p$

We had the theorem about $C_c^{\infty}$ being dense in $L^p$, which, as I understand, means that if we already have an $L^p$ function, there is a $C_c^{\infty}$ function arbitrary close to it with ...
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### Subsequence of functions in $L^p$

On a problem sheet we were asked to find a sequence of functions $(f_n)_{n \geqslant 0} \in L^p [0,1]$ such that $\lim_{n \to \infty} ||f_n||_p = 0$ but $\lim_{n \to \infty} f_n (x)$ doesn't exist ...
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### $L^p$ spaces and counting measure

currently I am working on the following two exercises as a revision for my exam. Let $\mu$ be the counting measure on $\mathbb N$. Show that if $1 \le p < s < \infty$ then $f \in L^p$ implies ...
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### Is there a sequence $(f_n)\in\ L^2([0,1])$, s.t. $\lVert f_n\rVert_2=1$, $\forall n$, but it has no convergent subsequences in $L^2([0,1])$ ?

Is there a sequence $(f_n)\in\ L^2([0,1])$, s.t. $\lVert f_n\rVert_2=1$, $\forall n$, but it has no convergent subsequences in $L^2([0,1])$ ? We know at least $(f_n)$, is not convergent in the ...
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### Suppose $1\le p < r < q < \infty$. Prove that $L^p\cap L^q \subset L^r$.

Suppose $1\le p < r < q < \infty$. Prove that $L^p\cap L^q \subset L^r$. So suppose $f\in L^p\cap L^q$. Then both $\int |f|^p d\mu$ and $\int|f|^q d\mu$ exist. For each $x$ in the domain ...
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### Function bounded a. e.

I have a question: if $f$ is uniformly bounded in $L^2(0,T,X)$ , then $f$ is uniformly bounded a.e. in $X \times (0,T).$ If yes, how to prove it? Thank you.
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### Lebesgue's convergence for $H(u_n)\nabla u_n$ where $H$ is not everywhere defined

Consider the Heaviside function that is undefined in zero, i.e. $$H(t)=\begin{cases} 1&t>0 \\ 0&t< 0\end{cases}$$ Now consider a sequence of $H^1(\Omega)$-functions $u_n\to u$ in the ...
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### Continuous embedding of $L^p$ into $L^q + L^\infty$

Let $L^p$ denote the Lebesgue-space over a $\sigma$-finite measure space $(\Omega,\mu)$. It is known that $L^{p_0} \cap L^{p_1} \hookrightarrow L^p \hookrightarrow L^{p_0} + L^{p_1}$ continuously ...
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### $L^\infty$ and the intersection of the spaces $L^p$

I'm trying to understand if it's true that: " if $f\in L^p\quad \forall p\in N\implies f\in L^\infty$"? My thoughts: Since $\int_R |f(x)|^p dx<\infty\quad\forall p\implies |f(x)|\to 0$? Can anyone ...
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### Is a $L^p$ function almost surely bounded a.e.?

I just have a quick question related to $L^p$ spaces. Any help is greatly appreciated. Is it true that if a function $f$ belongs to $L^p$ space, absolute value of $f$ raise to the power of $p$ is ...
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### $f(y-x)$ integrable implies $f=0$ a.e.

If $f(y-x)$ is in $L^p(\mathbb R^d\times\mathbb R^d)$, then I seem to conclude that $f=0$ a.e. (which seems wrong). My reasoning is that by Fubini and the integral's shift invariance (assume $p=1$ for ...
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### An almost orthogonality principle for $L^p$

If two functions are far from being orthogonal, their difference cannot be too large in $L^2$. A precise statement (easily verified with the Pythagorean theorem) is as follows: let ...
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### function in $L^1\setminus L^2$

I'm looking for an example of a function which belongs to the Banach space $L^1$ (i.e $\int|f|< \infty$) but is not in $L^2$ (so $\int|f|^2$ is unbounded). Does anyone know such a function?
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### Embedded Lp spaces [duplicate]

Let $L^\infty(Ω,F,P)$ be the vector space of bounded random variables $(X ∈ L^\infty (Ω,F,P)$ means that there exists a constant C such that $|X(ω)|≤C$, a.s.$)$. Show that ...
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### $L^{p}$ functions from Rudin Exercises 3.5

I am attempting a question from Rudin's "Real and Complex Analysis" Chapter 3 question 5. I shall summarise the question as below: Suppose that $f$ is a complex measurable function on $X$, $\mu$ a ...
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### Absolute Convergence of a Function

I have got stuck with a question. Please help me. Prove that $\dfrac{\sin(x)}{x}$ belongs to $L^p$ for all $p>1$. Thank You.
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### $L_p$ space,convergence

Let $1<p<\infty$ and $h\in L_p(\mathbb{R})$,that is,$\left(\displaystyle\int_{\mathbb{R}}|h|^p\right)^{1/p}<\infty$. Define a sequence $(f_n)_{n\in\mathbb{N}}$ by $f_n(x):=h(x-n)$. How to ...
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### Uniform convergence in $L^p$-spaces

Let $f\in L^p(0,\infty)$, $p>1$. Show that $\int_0^\infty f(x)\frac{\sin xy}{x} dx$ converges uniformly in $y$ in every finite interval. Show also that $|g(t+y)-g(y)|\leqslant M|t|^{\frac{1}{p}}$. ...
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### Why $f(x + x^3, y + y^3) \in L^1(\mathbb{R}^2)$, when $f(x, y) \in L^2(\mathbb{R}^2)$?

How show that $f(x + x^3, y + y^3) \in L^1(\mathbb{R}^2)$, when $f(x, y) \in L^2(\mathbb{R}^2)$? Can someone help me? Thank you!
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### Function in $L^1([0,1])$ that is not locally in any $L^{\infty}$

Can we find a function such that $f\in L^1([0,1])$ and for any $0\leq a<b\leq 1$ we have that $||f||_{L^{\infty}([a,b])}=\infty$?
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### How to show that a certain function is in $L^p$?

How can I show that the function $F(x)= \dfrac{|x| ^{-n+1}} { \log \frac{1}{|x|} }$, for $0 < |x| \leqslant \large\frac{1}{2}$ and $F(x)=0$, if $|x|>\large\frac{1}{2}$, is in ...
### Prove that $f(x)$ is integrable on $\mathbb{R}$.
Suppose $f(x)$,$xf(x)$ $\in$ $L_2(\mathbb{R})$. Prove that $f(x)\in$ $L_1(\mathbb{R})$.