$\vert T(x)\vert \leq \left\Vert T \right\Vert \left\Vert x \right\Vert$ 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.
Edit:
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.
