Is the notion of limit in Trotter Kurtz approximation theorem well defined? A Question concerning the paper of Kurtz (1969))

Question

In the first pages of the 1969 article "Extensions of Trotter’s Operator Semigroup Approximation Theorems" of Kurtz one reads:

This concept of limits seems a bit strange and not well defined.

Take for instance $L = L_n = S^1$ the unitary circle in $\Bbb{R}^2$ with the induced euclidean norm $x \in S^1 \Rightarrow \|x\| = 1$ Now take $P_n e^{i\theta} = e^{i\theta/n}$ then $\lim \|P_n x \| = 1$ So for any $f_n \to e^0$ we have from the above definition that $f_n \to \{x \mid x \in S^1\}$. Since for any $x$ $\lim_n \|f_n - P_n x\| = 0 $

Note however that $P_n$ is not linear. So this is no example of non uniqueness of the above concept of convergence. Still, how do we prove that the above concept of limit is well defined?

Answer 1

If I understand your question correctly, you want to prove the following statement:

Let $f,g \in L$ and $f_n \in L_n$. If $\lim_{n \to \infty} f_n = f$ and $\lim_{n \to \infty} f_n = g$, then $f=g$.

Proof: Since $f-g \in L$, we have by the definition of $P_n$

$$\|f-g\| = \lim_{n \to \infty} \|P_n(f-g)\| = \lim_{n \to \infty} \|P_n f - P_n g\|.$$

Hence, by the triangle inequality,

$$\|f-g\| \leq \limsup_{n \to \infty} \left(\|P_n f-f_n\|+\|P_ng - f_n\| \right) = 0$$

where the last identity follows from the fact that $\lim_n f_n = f$ and $\lim_n f_n = g$.