$\vert T(x)\vert \leq \left\Vert T \right\Vert \left\Vert x \right\Vert$ is a frequently-used inequality. Here, $x\in X$, $X$ is a normed space. $T$ is a functional. But I don't know how to relate $\left\Vert \cdot \right\Vert$ with $\left\vert \cdot \right\vert$. I know that $$\left\Vert T \right\Vert= \sup_{\left\Vert x \right\Vert=1} \left\Vert T(x) \right\Vert = \sup_{\left\Vert x \right\Vert \neq0}\frac{\left\Vert T(x) \right\Vert}{\left\Vert x \right\Vert} $$ Next, then I wonder why $\left\Vert T(x) \right\Vert \leq \left\Vert T \right\Vert \left\Vert x \right\Vert$, which I use frequently as well. However, I can't see how to prove it from the definition of norm space, and it looks like Hölder's inequality without specifying the norm. Can some give me some help? Thanks a lot.


About the inequality $\vert T(x)\vert \leq \left\Vert T \right\Vert \left\Vert x \right\Vert$, is it because that: if $0<p<q<\infty$ and $E$ is measurable, then $L^q(E) \subset L^p(E)$, therefore, $L^q(E) \subset L^1(E)$. But it is true only for $L^p$-norm.

  • 4
    $\begingroup$ What does $\sup$ mean? $\endgroup$
    – user228113
    May 12, 2018 at 15:39
  • $\begingroup$ That wasn't him being ignorant, that was him helping you. $\endgroup$
    – user223391
    May 12, 2018 at 15:41
  • 2
    $\begingroup$ @G.Sassatelli is giving you a hint to understanding the inequality. $\endgroup$ May 12, 2018 at 15:41
  • $\begingroup$ The norm on $T$ is defined in such a way that this inequality has to be true. If there were some $x$ not satisfying it, then the value we picked for the $\sup$ wasn't really a $\sup$. $\endgroup$ May 12, 2018 at 15:44
  • $\begingroup$ Oh, indeed. The second inequality is directly from moving the scalar ||x||, thanks $\endgroup$
    – Reinherd
    May 12, 2018 at 15:46

1 Answer 1



The case where $x=0$ is trivial.

Observe that for $x\not=0$, $$ \|T(x)\|=\left\|\|x\|T\left(\frac{x}{\|x\|}\right)\right\|=\|x\|\left\|T\left(\frac{x}{\|x\|}\right)\right\|\leq\|x\|\sup_{\|y\|=1}\|T(y)\|=\|x\|\|T\|. $$

The sketch is that you need to understand/convince yourself where each equality/inequality comes from.


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