Show that for any $1<p<\infty$ the set $\{ f \in L^p(\mathbb{R}) \cap L^1(\mathbb{R})\}$ where $ \int_{\mathbb{R}} f=0$ is dense in $L^p(\mathbb{R})$.

score 1.2k · Answer 13 · 2026-04-13 16:46:28

1.2k

Views

$Tf = xf(x)$ is not compact in $L^2([0,1])$

Published on 13 Apr 2026 - 16:46

score 48 · Answer 14 · 2026-04-11 01:24:36

48

Views

Are $L^p$ topologies compatible?

Published on 11 Apr 2026 - 1:24

score 66 · Answer 15 · 2026-04-10 00:18:57

66

Views

How to lift the restriction of sigma-finite by transfinite induction in proving linear functionals separate of $L^\infty(\Omega)$?

Published on 10 Apr 2026 - 0:18

List Question

Weak formulation of a nonlinear problem with test functions in a dense subspace of $H_0^1$

continuous linear functional on $l^{\infty}$ space

$C_{c}^{\infty}(\Omega)$ is dense in $L^{\infty}(\Omega)$ with respect to the topology $\sigma(L^{\infty},L^{1})$

Holders Inequality: Suppose $\int_{0}^\infty x^{-2}|f|^5 dx < \infty$. Prove that $\lim_{t \to 0} t^{-\frac{6}{5}} \int_0^t f(x)dx = 0$

Show that there exists $ \lambda \ge 0$ such that $v=\lambda u$

Why is $\frac{1}{x^{1/p} (\ln(x)^2+1)}$ in $L^1$ but not in $L^p$ for any $p>1$

Lebesgue-integrability of derivatives

Prove series converge for almost every $x$

Prove that following are true for $\phi \in C_c^{\infty}(\mathbb{R}), \phi \ne 0$

some detail calculation on the proof of equivalence of norms

Show that $A$ and $A^C$ are both dense in $(\ell^2,\lVert \cdot \rVert_2)$, where $A=\{x\in\ell_2:\sum_{k=1}^\infty x_k\neq0\}$.

Show that for any $1<p<\infty$ the set $\{ f \in L^p(\mathbb{R}) \cap L^1(\mathbb{R})\}$ where $ \int_{\mathbb{R}} f=0$ is dense in $L^p(\mathbb{R})$.

$Tf = xf(x)$ is not compact in $L^2([0,1])$

Are $L^p$ topologies compatible?

How to lift the restriction of sigma-finite by transfinite induction in proving linear functionals separate of $L^\infty(\Omega)$?

Trending Questions

Popular # Hahtags

Popular Questions