Dear Mieczyslaw,

It is well known that the weak topology is Hausdorff. Also this topology is regular. Moreover, if $\mathcal{U}=\{U_s:s\in I\}$ is an open covering of a reflexive Banach space $X$, then for every $n\in N$,

$B[0,n]=\{x\in X:\|x\|\leq n\}\subseteq \cup U_s$.

By compactness of $B[0,n]$ in the weak topology, it follows that for every $n\in N$, there exist $\alpha^n_1,\dots,\alpha^n_{s_n}\in I$ such that

$B[0,n]\subseteq\cup\{ U_{\alpha^n_i}:i=1,\dots,s_n\}$.

Thus $\underset{n\in\mathbb{N}}{\cup}\{U_{\alpha^n_i}:1=1,\dots,s_n\}$ is a countable subcovering of $\mathcal{U}$.

Therefore $X$ is Lindelöf.

Finally, since every regular Lindelöf space is paracompact, it follows that if $X$ is a reflexive Banach space then $X$ with the weak topololy is paracompact.

Thank you very much! Several people said me, that the answer is known, but without any references. Do you know some spaces being examples or counterexamples for Theorem 1?

Yours,

      M.C.

It seems to me that the words "weak topology" in Morita's paper (in the reference given by George) are not related to the meaning of these words in Mieczyslaw Cichon question.

Thank you all! I see that my question, however, is not trivial.

   Sincerely,

     M.C.

Some more information from the following classical paper may also be useful:

 H. H. Corson-The weak topology of a Banach space, TAMS(1961), p 1-15.

http://www.ams.org/journals/tran/1961-101-01/S0002-9947-1961-0132375-5/S0002-9947-1961-0132375-5.pdf

Some extracts from page 8 of the paper (see also its Theorem 4, p. 12):

Now, recall that a topological space B is paracompact if each open cover of B has a locally finite refinement . ......It is known that a Lindelöf space is necessarily paracompact.

Lemma 5. (p. 8) If a Banach space B is w-paracompact, then B is w-Lindelof.

.........

The next few lemmas give an indication of how near to being w-paracompact certain B are. (See [11, Chapter 5] (Kelley) for definitions, as well as proofs that paracompact spaces have the properties referred to in these lemmas.).....

Thank you, Professor Liaqat Khan! It is exactly the paper which is a basis for my question. Unfortunately, in some future applications, I need precisely "paracompactness" and the current problem is to find either class of Banach spaces or family of subsets of a Banach space in which the weak topology is paracompact. Thank you for your recommendation!

Dear Mieczyslaw,

It is well known that the weak topology is Hausdorff. Also this topology is regular. Moreover, if $\mathcal{U}=\{U_s:s\in I\}$ is an open covering of a reflexive Banach space $X$, then for every $n\in N$,

$B[0,n]=\{x\in X:\|x\|\leq n\}\subseteq \cup U_s$.

By compactness of $B[0,n]$ in the weak topology, it follows that for every $n\in N$, there exist $\alpha^n_1,\dots,\alpha^n_{s_n}\in I$ such that

$B[0,n]\subseteq\cup\{ U_{\alpha^n_i}:i=1,\dots,s_n\}$.

Thus $\underset{n\in\mathbb{N}}{\cup}\{U_{\alpha^n_i}:1=1,\dots,s_n\}$ is a countable subcovering of $\mathcal{U}$.

Therefore $X$ is Lindelöf.

Finally, since every regular Lindelöf space is paracompact, it follows that if $X$ is a reflexive Banach space then $X$ with the weak topololy is paracompact.