# Tagged Questions

For questions about $L^p$ spaces, that is, given a measure space $(X,\mathcal F,\mu)$, the vector space of equivalence class of measurable functions with $p$-th power of the absolute value integrable. Question can be about properties of elements of these spaces, or when the ambient space on a ...

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### Is it natural how $L^p$ spaces measure local and global sizes the same?

This is a continuation of my question Spaces of functions similar to $L^p$ but with different local and global sizes. I have been bothered by the fact that the $L^p$ norm on $\mathbb R^n$, which is ...
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### Spaces of functions similar to $L^p$ but with different local and global sizes

Obviously much of analysis on $\mathbb R^n$ considers $L^p$ spaces or other Banach spaces derived from them. The definition of $L^p(\mathbb R^n)$ looks very natural, but I've been bothered for some ...
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### Inclusion of Schwartz space on $L^p$

I'm looking for a proof of $\mathcal{S}(\mathbb{R}) \subset L^p(\mathbb{R})$ for $1 \leq p \leq \infty$. My informal probe follow like this: For any function $f \in L^p(\mathbb{R})$ exists a ...
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### Is the sequence of functions $g_n=ng_1(nx)$ a Cauchy sequence?

Given a function $g_1(x) \in \mathcal L^2(\Bbb R)$ that satisfies: $$\int_{-\infty}^{\infty}dx \space g_1(x)=1$$ one can define a sequence of functions $g_n=ng_1(nx)$. Does $g_n(x)$ define a ...
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### Easy example of $f\in L_1^*\backslash L_\infty$?

If I'm not mistaken the dual of $L_1$ is $L_\infty$ whenever the measured space is $\sigma$-finite. So I know where not to look for an easy example of $f\in L_1^*\backslash L_\infty$. Does anyone know ...
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### Definition of $L^p(C,B)$ where $C$ is a subset of the domain of the function and $B$ is a Banach space.

I am looking for the definition of $L^p(C,B)$ as declared in the title. I know that $C^s(C,B)$ is defined with the norm: $$\frac{ \| f(x) - f(y)\|_B}{|x-y|^s}\le A$$ i.e $f\in C^s(C,B)$ iff the ...
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### In Rudin's proof of the completeness of $L^\infty$

This is closely related to a previous question: In a proof of the completeness of $L^\infty$ The following is a proof of completeness of $L^\infty$ by Rudin in his Real and Complex Analysis: Here ...