# what is the difference between $||T_nx||$ and $||T_n|| ?$

In the uniform Boundedness theorem it is written $$||T_nx|| \le c_x$$ and $$||T_n|| \le c$$

My confusion : what is the difference between $$||T_nx||$$ and $$||T_n||?$$

My thinking : In real analysis we take $$f_n(x)=f_n$$ so here im confusing that

Are both $$||T_nx||$$ and $$||T_n||$$ are same?

• $f_n(x) = f_n$ is not true. One is a real number, the other is a function. Apr 28, 2021 at 15:18

They are not the same. $$\|T_n\|$$ is the operator norm of $$T_n$$, but $$\|T_n x\|$$ is the norm of the specific vector $$T_n(x)$$, where $$x$$ is an element of the domain of $$T_n$$.
$$\|T_n\|=\sup_{\|x\|\leq 1} \|T_n(x)\|$$
of some $$T_n$$.
$$\|T_n x\|$$ is the norm of the image $$T_n(x)$$.
In other words, the statement $$\| T_n x \| \le c_x$$ reads $$\forall \ x \in E \quad \exists \ c \ge 0 \quad \forall \ n \in \mathbb{N} \quad \| T_n x \| \le c,$$ while the statement $$\| T_n \| \le c$$ is equivalent to $$\exists \ c \ge 0 \quad \forall \ x \in E \quad \forall \ n \in \mathbb{N} \quad \| T_n x \| \le c.$$ The latter is obviously stronger. However, Banach-Steinhaus theorem shows they are actually equivalent if $$E$$ is a Banach space.
And of course, $$\| \cdot \|$$ here means two different things, as Ben Grossmann pointed out.