1
vote
1answer
55 views

Inequality for norm of linear combination of linearly independent vectors

I'm trying to find a proof for the following: Let {$u_{1},...,u_{n}$} be a linearly independent set of a normed space $X$. Then, there is a constant $c>0$ such that for every set of scalars ...
3
votes
1answer
149 views

Equality in Young's inequality for convolution

I am trying to understand how sharp Young's inequality for convolution is. The inequality says $||f \ast g||_r \leq ||f||_p ||g||_q$ where as $1/p+1/q = 1+1/r$. Actually, there are a couple of papers ...
1
vote
2answers
64 views

Inequality regarding norm vector space

I am not sure how to prove this inequality involving norms. Let $X$ be a normed vector space and $x,y$ are vectors in $X$ with nonzero norms. Prove the following inequality is true. $$\|x-y\|\geq ...
2
votes
1answer
56 views

Endpoint-average inequality for a line segment in a normed space

Let $X$ be a normed vector space over $\mathbb R$. What is the smallest universal constant $C>0$ such that the inequality $$\|x\|\le C\int_0^1 \|x+tv\|\,dt\tag{1}$$ holds for all $x,v\in X$? ...
1
vote
2answers
144 views

Inequality involving Lp space, Holder Space, Sobolev Space

Do we have the following inequality or some variation of this: $||u||_{L^{p}(U)} \leq ||u||_{C^{0,\gamma}(\bar{U})} \leq ||u||_{W^{1,p}(U)}$ for $n < p \leq \infty$ for $u \in W^{1,p}(U)$ where $U$ ...
4
votes
2answers
47 views

Minkowski type inequality in Banach algebras

Under which circumstances it is true that $\|(A+B)^n\|^{1/n}\le \|A^n\|^{1/n}+\|B^n\|^{1/n}$ for elements $A$ and $B$ in a Banach algebra and a natural number $n$?
0
votes
0answers
53 views

Explanation on the proof of the continuous Hardy inequality

Here there is a proof of the continuous Hardy inequality (theorem 2). I would like a explanation on the following passage. ...
1
vote
1answer
124 views

How to show Zygmund space is Hölder space?

The motivation of this question is to show that Zygmund space is Hölder space, in certain cases. For simplicity, take $s\in (0,1)$, I want to show $$\|f\| = \|f\|_\infty + \sup_{x,y\in ...
2
votes
1answer
137 views

Inequality for norms

Let g(x, y) be function on $X\times Y$. Show that for all $p\geq q$ $$ \|\,\|g\|_{L^q(Y)}\,\|_{L^p(X)}\leq \|\,\|g\|_{L^p(X)}\,\|_{L^q(Y)} $$ Thsnk you.
4
votes
0answers
72 views

Find Lipshitz constant

Let $x, y$ be two vectors in $R^n$. Let $\pi$ be permutation on $\{1, \ldots, n\}$ with uniform distribution. Find Lipshitz constaants of the following functions: $$ f(\pi)=\sum_{i=1}^nx_{\pi(i)}y_i ...
3
votes
2answers
199 views

Two proofs concerning Hölder's inequality

I am studying functional analysis and I have come across two statements which can be proven by using Hölder's inequality, but I don't know how/why. Reminder: let $a\in l^p$ and $b\in l^q$, then: $$ ...
1
vote
2answers
64 views

There does not exist $c \ge 0$ such that $\|f\|_{\max} \le c\|f\|_1$

Let $f \in C[a,b]$ and let $\|f\|_1$ be the $\mathcal{L}^{1}$-norm and $\|f\|_{\max} = \max_{x \in [a,b]}|f|$. They are both norms on the given vector space. I want to prove that $\not \exists c \ge ...
5
votes
2answers
376 views

Is there a lower-bound version of the triangle inequality for more than two terms?

The triangle inequality $|x+y|\leq|x|+|y|$ can be generalized by induction to $$|x_1+\ldots+ x_n|\leq|x_1|+\ldots+|x_n|.$$ Can we generalize the version $|x+y|\geq||x|-|y||$ to $n$ terms too? I need ...
5
votes
1answer
348 views

Norm in a dual space

If $f \in X^*$, with $X^*$ the dual space consisting of all linear bounded functionals on a linear normed space $X$. With the norm defined as $||f||_{X^{*}} = \sup_{||x|| \leqslant 1} |f(x)|$. Why ...
1
vote
1answer
214 views

Poincaré inequality using $H^1$ seminorm

Does this inequality holds for Poincaré Inequality? $$||v||_{L^2} \leqslant C_p |v|_{H^1}$$ and $$ |v|_{H^1} = ||v'||_{L^2} $$ where $| \dot~ |$ is the semi norm and $||\dot~||$ is the norm. I'm ...
1
vote
2answers
123 views

My “wrong” comparison between $\ell^2$ and $\ell^1$

For sure, $\ell^2$ is larger than $\ell^1$, because for $|x|<1$, $|x|^2<|x|,$ that is, $||x||_2\leq||x||_1.$ But using Cauchy-Schwartz inequality, I get a "wrong" comparison: ...
3
votes
1answer
138 views

Prove $\|F(x,·)\|_{L^{4,1}(E)}^2 \leq C\|F(x,·)\|_{L^{\infty}(E)}\|F(x,·)\|_{L^{2,2}(E)}$

How to derive this inequality? $$\|F(x,·)\|_{L^{4,1}(E)}^2 \leq C\|F(x,·)\|_{L^{\infty}(E)}\|F(x,·)\|_{L^{2,2}(E)},$$ where $C$ is constant and ...
7
votes
1answer
1k views

Inequalities in $l_p$ norm

I'm having difficulty with the following problem. Any help would be appreciated. Problem: Consider the sequence spaces $l_p$ with the usual norm. If $1\le p\le q\le \infty$, I want to show the ...
1
vote
0answers
326 views

Convergence of $L^p$ norms

Given a measure space $X$ with its measure $\mu$, it can be shown (I'll provide a proof if asked for) that $\displaystyle \forall f \in L^\infty(X,\mu),~\textrm{if } \exists p_0:\forall q \geq p_0, ...
2
votes
3answers
748 views

Cauchy-Schwarz Inequality

In Luenberger book Cauchy-Schwarz Inequality is defined like this: For all $x,y$ in an inner product space $|(x|y)| \le \|x\|\|y\|$. Equality holds if and only if $x = \lambda y$ or $y = \theta$. ...