# 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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### If $f^p\in L^1([0,1])$ it's bounded a.e.

We know that being Lebesgue integrable does not imply boundedness of the function (e.g. $g(x)=\frac{1}{\sqrt x}$). However function in $L^p$ spaces are functions with some decay conditions. Suppose ...
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### Linear operator on a dense subset of $L_p$ which is unbounded when extended to $L_p$

Let the linear operator $L:C^{\infty}_0([-1,1]) \rightarrow C(\{0\})$ be defined as $f(0)$ (evaluating a function in $C^{\infty}_0([-1,1])$ at $0.$ I would like to show that extending this to ...
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### If $f_n + g_n \to h$ in $L^2(\Omega)$ and $f_n, g_n \geq 0$, does $f_n \to f$ for some $f$?

On a bounded domain $\Omega$, if $f_n + g_n \to h$ in $L^2(\Omega)$ and each $f_n, g_n \geq 0$, does $f_n \to f$ for some $f$? I feel like this should be true since each sequence is non-negative, so ...
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### Counterexample for $L^p$ Inclusion

The general question is : disprove that $L^p(\mathbb R)\subset L^q(\mathbb R)$ and $L^q(\mathbb R)\subset L^p(\mathbb R)$ for $q<p\leq\infty$ I managed to find counterexamples for the finite cases ...
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### $L^p \subset L^q$ for $p\neq q$.

Let $1\leq p \leq q \leq \infty$. It's well known that on a finite measure space $(X,\mathcal{M}, \mu)$, we have the inclusion $L^q(X,\mathcal{M}, \mu) \subset L^p(X,\mathcal{M}, \mu)$. Questions ...
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### Showing that the following set is closed in $L^p$
Question: Let $1\le p < \infty$ and $1\le q \le \infty$. Prove that the following set is closed in $L^p$. $$\{f \in L^p \cap L^q : |f|_q \le 1 \}$$ My try: Let $f_n$ be a sequence in the above ...
### Is $D(A)$ necessarily dense in $E$? Is $G(A)$ necessarily closed in $E \times E$?
Let $E = L^p(0, 1)$ with $1 \le p < \infty$. Consider the unbounded operator $A: D(A) \subset E \to E$ defined by$$D(A) = \{u \in W^{1, p}(0, 1),\text{ }u(0) = 0\} \text{ and }Au = u'.$$I have two ...