Questions about If X is reflexive, X is weakly sequentially complete in Conway's book.

Question

I don't understand this proof. Which PUB is using here and why the $\lim\langle x_n,x^*\rangle$ exists from $x_n$ clusters to $x$?

Thank you!

Answer 1

From the definition of weakly Cauchy, we know that for each $x^*$ in $\mathscr{X}^*$ the sequence $(\langle x^*,x_n \rangle)_n$ is Cauchy in $\mathbb{F}$. Hence $(\langle x_n,x^* \rangle)_n$ is Cauchy in $\mathbb{F}$ for each $x^*\in\mathscr{X}^*$, where the $x_n$ here is viewed as the image of the element $x_n$ under the canonical isometry from $\mathscr{X}$ into $\mathscr{X}^{**}$. Since Cauchy sequences are bounded, we have $$ \sup_n |\langle x_n,x^* \rangle| < \infty $$ for every $x^*\in\mathscr{X}^*$. As the space $\mathscr{X}^*$ is complete, we can apply the uniform boundedness principle to obtain that $\sup_n\|x_n\|<\infty$. Taking $M:=\sup_n\|x_n\|$, we have the $M$ used in Conway's proof. Then the argument proceeds, as Conway writes, by using the fact that the unit ball of a reflexive space is weakly compact.

The limit $\lim_n \langle x_n,x^* \rangle$ exists because we know $(\langle x_n,x^*\rangle)_n$ is Cauchy in $\mathbb{F}$ and $\mathbb{F}$ is complete.