How does weak convergence implies strong convergence in the setting of compact inclusion?

Asked Oct 16 '16 at 12:34

Active Oct 21 '16 at 13:31

Viewed 157 times

The following hand-waving argument is in the proof of Lemma 8.4 in Constantin and Foias's Navier-Stokes Equations:

Suppose $X,Y,Z$ are Hilbert spaces such that we have the following embedding properties $$ X\overset{\text{cpt}}{\hookrightarrow}Y\hookrightarrow Z. $$ ($X$ is compactly embedded in $Y$, and $Y$ is continuously embedded in $Z$.)

Suppose $x_n\to 0$ weakly in $X$. Then $x_n$ convergens strongly in $Y$ and a fortiori in $Z$.

Would anyone explain why this is true (or I misses something that this argument needs more assumptions)? What does "a fortiori" mean here?

edited Oct 21 '16 at 13:31

asked Oct 16 '16 at 12:34

1

To answer the last question, a fortiori can be understood as thus necessarily or hence certainly. – Brian M. Scott Oct 16 '16 at 12:37
1

Should it be strongly in $Y$? – Anthony Carapetis Oct 16 '16 at 12:39
@AnthonyCarapetis: yes, thanks! – Oct 16 '16 at 12:39
@BrianM.Scott: Thanks for you comment! Maybe I should put a topology tag to this question I think? – Oct 16 '16 at 12:41
@Jack: That should be okay, yes. – Brian M. Scott Oct 16 '16 at 12:42

How does weak convergence implies strong convergence in the setting of compact inclusion?

0 Answers0