# Tagged Questions

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### 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 ...
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### 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 ...
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### 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.
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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 $\not \exists c \ge ... 2answers 376 views ### Is there a lower-bound version of the triangle inequality for more than two terms? The triangle inequality$|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 ... 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 ... 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 ... 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: ... 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 ... 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 ... 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, ...
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 $y = \theta$. ...