Show that if $\int fh < \infty$ for all $h \in L^q$ then $f \in L^p$
24777
If $f$ is measurable and $fg$ is in $L^1$ for all $g \in L^q$, must $f \in L^p$?
148582
Proof: $\|X\|_2\leq \sqrt{n}\|X\|_\infty$
108816
$L^p$ integrable but not $L^q$ integrable
20467
Show that $(y_n)\in l^q$
129331
Prove that $|x^p - y^p| \le p|x-y|(x^{p-1} + y^{p-1})$ provided that $1 \le p \lt \infty$ and $x, y \ge 0$
157455
Show that the norm of the multiplication operator $M_f$ on $L^2[0,1]$ is $\|f\|_\infty$
81302
Showing that $a_n \in \ell^{\infty}$
42108
Prove that $c_0$ is closed in $\ell^\infty$
41057
Does $L^p$-convergence imply pointwise convergence for $C_0^\infty$ functions?
29114
If $f \in L^{p_1}(E) $ is bounded then $f\in L^{p_2}$ for any $p_2>p_1$
9613
If $f_k \to f$ a.e. and the $L^p$ norms converge, then $f_k \to f$ in $L^p$
41173
If $f\in S_\infty$ and $\int_{\mathbb{R}}x^pf(x)d\mu=0$ for all $p\in\mathbb{N}$ then $f\equiv 0$?
135700
Show that $\lim_{n \to \infty} (1/n)\sum_{i=1}^n f(x_i)$ exists
12865
Prove that $\lim_{n\to\infty} H_n/n = 0$ ($H_n$ is the $n$-th harmonic number) using certain techniques
95921
Question about $\lim _{q \rightarrow \infty}\|f\|_{q}=\|f\|_{\infty}$
9159
$L^p$-norm of a non-negative measurable function
142025
How can I show that if $f\in L^p(a, b)$ then $\lim_{t\to 0^{+}}\int_{a}^b |f(x+t)-f(x)|^p\ dx=0$..