query: Show that if $f$ is a bounded function on $E$ and $f\in L^{p_1}$ then $f\in L^{p_2}$ for any $p_2>p_1$.
positive_passages.docid: 29114
positive_passages.text: If $f \in L^{p_1}(E) $ is bounded then $f\in L^{p_2}$ for any $p_2>p_1$
negative_passages.docid: 150896
negative_passages.text: Show that $\lim_{p \to \infty} \| f \|_p = \| f \|_\infty$
negative_passages.docid: 10383
negative_passages.text: The $ l^{\infty} $-norm is equal to the limit of the $ l^{p} $-norms.
negative_passages.docid: 106401
negative_passages.text: Prove: $p-mq | f(m)$ where $m$ is any integer
negative_passages.docid: 71857
negative_passages.text: Proving that, $|f'(x)-f'(y)|\le k|x-y| \implies (f'(x))^2< 2 kf(x) $
negative_passages.docid: 6450
negative_passages.text: Show $\mathbb{E}[f(X)g(X)] \geq \mathbb{E}[f(X)]\mathbb{E}[g(X)]$ for $f,g$ bounded, nondecreasing
negative_passages.docid: 74202
negative_passages.text: Show that $|f(z_1)|=|f(z_2)|≤|z_1|^2=|z_2|^2$.
negative_passages.docid: 95626
negative_passages.text: $\int_X |f_n - f| \,dm \leq \frac{1}{n^2}$ for all $n \geq 1$ $\implies$ $f_n \rightarrow f$ a.e.
negative_passages.docid: 63124
negative_passages.text: Show that $ \lVert A \rVert_2^2 \leq \lVert A \rVert _1 \lVert A \rVert _ \infty $
negative_passages.docid: 157455
negative_passages.text: Show that the norm of the multiplication operator $M_f$ on $L^2[0,1]$ is $\|f\|_\infty$
negative_passages.docid: 17314
negative_passages.text: Use property to show that if $f_n \rightarrow f$ in measure on finite measure set $E$ then $f_n^2 \rightarrow f^2$ in measure on $E$.
negative_passages.docid: 113411
negative_passages.text: If $E$ is generated by $E_1$ and $E_2$, then $[E:F]\leq [E_1:F][E_2:F]$?
negative_passages.docid: 43897
negative_passages.text: An integral inequality (Rudin $L^p$-spaces 3.12)
negative_passages.docid: 52249
negative_passages.text: Is it true that $||a|^p - |b|^p| \le C_p|a - b|^p$ for $1 < p < \infty$?
negative_passages.docid: 9613
negative_passages.text: If $f_k \to f$ a.e. and the $L^p$ norms converge, then $f_k \to f$ in $L^p$
negative_passages.docid: 91352
negative_passages.text: If $\Re(f)$ is bounded then f is constant.
negative_passages.docid: 158647
negative_passages.text: If $|f(z)|\le |f(z^2)|$ then prove that $f$ is constant