1

Let $X = L^2.$

Ssuppose I have $f_n \to f$ in $X^*$ and $T_\epsilon x_n \to T_\epsilon x$ in $X$. Here $T_\epsilon$ is a continuous map from $X$ into itself. So $$\langle f_n, T_\epsilon x_n \rangle \to \langle f, T_\epsilon x \rangle$$ holds as $n \to \infty$.

Suppose that $\lim_{\epsilon \to 0}\langle f_n, T_\epsilon x_n \rangle $ exists. When I can deduce that $$\lim_{\epsilon \to 0}\langle f_n, T_\epsilon x_n \rangle \to \lim_{\epsilon \to 0}\langle f, T_\epsilon x \rangle$$ as $n \to \infty$?

maths_student_2000
  • 85
  • 6

1 Answers1

2

Let's put $\Phi(n,\epsilon) = \langle f_n, T_\epsilon x_n\rangle$. You are asking for a sufficient condition for $$\lim_{n\to\infty} \lim_{\epsilon\to 0} \Phi(n,\epsilon) = \lim_{\epsilon\to 0} \lim_{n\to\infty}\Phi(n,\epsilon) $$ A standard sufficient condition is uniformity of one of these limits, but there are others. See:

Community
  • 1