Sequences of Continuous Linear Operators between Banach Spaces.

Let $$E, F$$ be two Banach Spaces. Let $$\{ T_{n} \}$$ be a sequence of continuous linear operators from $$E$$ into $$F$$ such that:

For all $$x \in E: T_{n}x \rightarrow Tx,$$ some limit in $$F$$.

Then the following 3 properties are satisfied:

(a) $$^{\text{sup}}_{n \in \mathbb{N}} ||T_{n}|| < \infty$$

(b) $$T \in L(E,F)$$

(c) $$||T|| \leq ^{\text{lim inf}}_{n \rightarrow \infty} ||T_{n}||$$

Here $$L(E,F)$$ is the space of continuous linear functions from $$E$$ to $$F$$ with the norm $$|| T || = \ ^{\ \text{ sup}}_{|x| \leq 1} |Tx|$$.

I have proven properties (a) and (b) easily, but (c) is a problem. I am under the assumption that the author of the textbook I am reading has written the property this way because $$^{\text{lim}}_{n \rightarrow \infty} ||T_{n}||$$ does not necessarily exist, since if it did, the lim sup would just equal the limit.

I know that $$|T_{n}x| \leq ||T_{n}|| \cdot |x|,$$ for all $$n \in \mathbb{N}, x \in E$$, but am unable to see how I can use this without a limit for $$||T_{n}||$$.

• Why not take liminf both sides and by continuity of absolute value we get $|Tx| \leq \liminf\limits_{n\to \infty} ||T_n|| |x|$? Now it follows by definition. – James Yang May 11 at 14:50

Fix $$x\in E$$ or norm one: then $$\left\vert Tx\right\rvert_F=\lim_{n\to +\infty}\left\vert T_nx\right\rvert_F$$ hence taking the $$\liminf_n$$ in the inequality $$\left\vert T_nx\right\rvert_F\leqslant \lVert T_n\rVert$$, we get that $$\left\vert Tx\right\rvert_F\leqslant \liminf _{n\to +\infty}\lVert T_n\rVert .$$ Since this estimate is valid for all $$x$$ of norm one, we see that the operator norm of $$T$$ does not exceed $$\liminf _{n\to +\infty}\lVert T_n\rVert$$.