# 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 ...

42 views

### Approximate Sobolev function by smooth function - error estimate?

I wondered if there is, in general, a way to estimate the error (in $L^2$) between a Sobolev function and its mollified version. Let's say $\Omega\subset\mathbb{R}^n$ is bounded with smooth boundary ...
14 views

38 views

### Intuitive explanation of p-norm in finite and infinite dimensinos

I am not a mathematician, so very rigorous treatment with things that only a math major learns will not suffice here. I want to learn about p-norms and i can't quite get the intuition behind them. I ...
112 views

### Integral operator is bounded on $L^p$ if it maps $L^p$ to itself

Here is a homework excercise. Let $(X,\Omega,\mu)$ be a $\sigma$-finite measure space,$1\leq p <\infty.$ and suppose that $k:X\times X\rightarrow \mathbb{C}$ is an $\Omega \times \Omega$ ...
26 views

### subspace of $L^2$

Let $(X,B(X),\mu)$ a measurable space, for a positive finite measure $\mu$, we consider $H=L^2(X,d\mu)$, Let $A$ a closed subspace of $H$, we know that $A$ is a hilbert space, can we say that it exist ...
53 views

19 views

### Compactness Sobolev embedding for even functions on $\mathbb{R}$.

It is well-known from Lions's article,"Symétrie et compacité dans les espaces de Sobolev", that the subspace $H^s_r(\mathbb{R}^n)$ of the Sobolev space $H^s(\mathbb{R}^n)$ containing all radial ...
33 views

### $L^{\infty}$ convergence for random variable

I am slightly confused with this borderline case regarding $L^p$ convergence. In some probability books, they clearly state that $p<\infty$ whereas the online sources do not impose this ...
59 views

50 views

### Showing that a subspace of $L^p$ is closed

I would like to prove that a particular subspace of $L^p([1,\infty[)$ (for some $p \in [1,\infty[$) is closed, but I'm not sure how to do it properly. For any sequence $(x_n) \in \ell^p$, let ...
51 views

27 views

### 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 ...
23 views

### 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 ...
39 views

34 views

25 views

38 views

### 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 ...
### $L^p \subset L^q$ 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 ...