Im looking for an element in $\ell^p$ which is not in $\ell^q$ for all $1\leq q&lt;p$. Does anybody know such a series?
121959
For every $p\in [1,\infty)$, consider the set $\ell^p$ of real sequences $x=(x_n)=(x_1,x_2,\ldots)$ such that $\sum_{n=1}^\infty |x_n|^p&lt;\infty$. This is a Banach space with the norm $\Vert x\Vert_p=(\sum|x_n|^p)^{1/p}$.  I have the following problem:     Is it true that $\ell^p=\bigcup_{1\leq q&lt;p}\ell^q$ for $p&gt;1$?   I don't think it is true, but I'm having problems to show this. Clearly, the statement is true for some $p&gt;1$ iff it is true for every $p&gt;1$.  I'm trying to find a sequence $(a_n)$ such that $\sum |a_n|^s$ converges iff $s\geq 1$. I've tried many sequences (such as $(n^{1+1/n})$, involving $\log$, etc...), but none of them worked.
93071
Let $p,q,r \in (1,\infty)$ with $1/p+1/q+1/r=1$. Prove that for every functions   $f \in L^p(\mathbb{R})$, $g \in L^q(\mathbb{R})$,and $h \in L^r(\mathbb{R})$    $$\int_{\mathbb{R}} |fgh|\leq \|f\|_p\centerdot \|g\|_q \centerdot\|h\|_r.$$   I was going to use Hölder's inequality by letting  $1/p+1/q= 1/(pq/p+q)$ and WLOG  let $p&lt;q$ so that $L_q(\mathbb{R})\subseteq L_p(\mathbb{R})$, but I cannot use this inclusion because $\mathbb{R}$ does not have finite measure.   Would you please help me if you have any other method to approach this problem?
51418
Let $\Bbb{P}$ be the set of prime numbers and $Q=\Bbb{P}\cup\{1\}$. Is it true that every natural numbers ($\neq 0$) is the sum of distinct elements of $Q$? I tried from $1$ to $60$ and it seems true.
98160
This is a small exercise that I just can't seem to figure out. When I see it I'll probably go 'ahhh!', but so far I haven't made any progress.  I'd like to prove that any linear functional $\phi$ on $\ell^\infty$ that is positive, i.e. for any $(x_n) \in \ell^\infty$ such that $x_n \geq 0$ for all $n \in \mathbb{N}$ we have $\phi((x_n)) \geq 0$, is a bounded linear operator and has $\|\phi \| = \phi((1,1,...))$.  Nothing I have tried worked so far, so any hints would be appreciated, thanks!
56312
Can someone help me? I can't even understand what the question is asking for.     Let $S = \{x_1,x_2,x_3,\ldots,x_n\}$. Show that ${\sim}[\forall x\in S,P(x)\lor Q(x)]\iff \exists x\in S,[{\sim}P(x)\land {\sim}Q(x)]$
82941
I'd appreciate your help showing that if $p\equiv3\,\left(\mbox{mod 4}\right)$   then p can't be written as a sum of two squares.  Thanks!
124249
I need to know if every group whose order is a power of a prime $p$ contains an element of order $p$? Should I proceed by picking an element $g$ of the group and proving that there is an element in $\langle g \rangle$ that has order $p$?
116544
I have done part $a$ and $b$, stuck on $c$. I don't know what to do here. I know that since $G$ has order $p^2$, any  element must have order $1, p$ or $p^2$ so any generator must have one of those orders, but I don't know how to reason any further..
23973
Q: For $p\in\mathbb{R}$ such that $2&lt;p&lt;\infty$, and $a,b&gt;0$, prove the following  $$a^p+b^p&lt;(a^2+b^2)^{p/2}$$  My attempt: I say it's equivalent to proving that $(a^p+b^p)^{1/p}&lt;(a^2+b^2)^{1/2}$. Then I did binomial expansions on both sides of the inequality. Both the expansions have $a+b$ so I canceled them. For the rest of the terms, I thought I'd get geometric series on both sides and show that the series on the left is smaller than that of the right by using related sum formulas. But what I found were not quite geometric. For the left side  $$\frac{1}{p}a^{1-p}b^p+\frac{1-p}{2p^2}a^{1-2p}b^{2p}+\frac{(1-p)(1-2p)}{6p^3}a^{1-3p}b^{3p}+\ldots$$  While for the right  $$\frac{1}{2}a^{-1}b^2-\frac{1}{8}a^{-3}b^4+\frac{1}{16}a^{-5}b^6+\ldots$$  How to proceed to prove the inequality?
20622
I can't seem to work out the inequality $(\sum |x_n|^q)^{1/q} \leq (\sum |x_n|^p)^{1/p}$ for $p \leq q$ (which I'm assuming is the way to go about it).
75052
Let $(a_n)_{n}\in\ell^2:=\ell^2(\mathbb{R})$ be a fixed sequence. Consider the subspace $$C=\{(x_n)_{n}\in\ell^2 : |x_n|\le a_n\text{ for all }n\in\mathbb{N}\}.$$  According to the book [Dunford and Schwartz, Linear operators part I, page 453] $C$ is compact in the $\ell^2$-norm, but there is no proof. How can I show that $C$ is indeed compact in $\ell^2$ ?
86253
Prove that $n&lt;3^n$ where $n \in \mathbb N$,  when $ n=1 $, I have proved it's true. And assumed when $n = p$ , $p&lt;3^p$ is true. Can any body help me in showing that it is true for $n =p+1$
14225
Let $\ell$ be an even positive integer.  Express $$\sum_{k=0}^n\sum_{i=0}^\ell (-1)^i\binom{n}{k}^2\binom{2k}{i}\binom{2n-2k}{\ell-i}$$in closed form.   Is anybody to give a insight, please? [A Path to Combinatorics for Undergraduates] p.192 problem 8.9
67021
Suppose $f\in L^{p}(\mathbb{R}^{n}) \cap L^{q}(\mathbb{R}^{n})$. How can I prove that for any $p \lt r \lt q$,  $$ \lVert f \rVert_{r} \leq (\lVert f \rVert_{p})^{(1/r-1/q)/(1/p-1/q)} (\lVert f \rVert_{q})^{(1/r-1/q)/(1/p-1/q)}\:? $$ I tried using Hölder to show that $f$ is in $L^{r}$ but I'm completely lost on how to go about it...
38967
Suppose $1&lt;p_1&lt;p_2&lt;\infty$, then show $L^{p_1}[a,b] \supset L^{p_2}[a,b]$.  I was able to show $\|f\|_{p_1} \le \|f\|_{p_2} (b-a)^{1/p_1 - 1/p_2}$ but I'm not sure how to proceed from here.
115817
I am curious about whether $|x|^p$ with $0&lt;p&lt;1$ satisfy $|x+y|^p\leq|x|^p+|y|^p$ for $x,y\in\mathbb{R}$.   So far my trials show that this seems to be right... So can anybody confirm whether this is right or wrong and give a proof or counterexample?
113062
Another exponent problem.  $GL(n,q)$ is the group of invertible $n\times n$ matrices over the finite field $GF(q)$, where $q$ is a prime power.  I am trying to figure out the exponent of this group.  Ideas:  We'd be doin real good if we could find the set of element orders $\{o(A):A\in GL(n,q)\}$, so I am looking for things to help me figure this out.   The order of $GL(n,q)$ is $$|GL(n,q)|=\prod_{k=0}^{n-1}\left(q^n-q^k\right)$$ The set of prime factors of this number is a little unpredictable so I'm not sure whether I can use this. If $q$ is prime, the maximum order of an element in $GL(n,q)$ is at most $p^n-1$ (and this bound is strict I think).  Also I guess the maximal power of $p$ that occurs as an order of this group is $p^{\lceil \log_pn\rceil}$.  () I am aware of , which I believe implies that $\text{Exp}(GL(2,p^n))=\text{lcm}\{p(p^{2n}-1),p-1\}$.  I am trying to think of how to extend it to the general case but I'm not sure.   And I'm stuck; can't think of anything else that might help.