next up previous
Next: Proof of the main Up: Separable Banach space theory Previous: Trees and subspaces of

   
The weak-* topology in subsystems of Z2

The language of second order arithmetic consists of number variables m, n, ..., set variables X, Y, ..., primitives +, $\cdot$, 0, 1, =, $\in$, and logical operations including number quantifiers and set quantifiers. By second order arithmetic (sometimes called Z2) we mean the theory consisting of classical logic plus certain basic arithmetical axioms plus the induction scheme

\begin{displaymath}(\varphi(0)\wedge\forall
n(\varphi(n)\rightarrow\varphi(n+1)))\rightarrow\forall n\varphi(n)\end{displaymath}

plus the comprehension scheme

\begin{displaymath}\exists W\, \forall n\,(n\in
W\leftrightarrow\varphi(n))\end{displaymath}

where $\varphi(n)$ is an arbitrary formula of the language of second order arithmetic. In the comprehension scheme it is assumed that the set variable W does not occur freely in $\varphi(n)$. All of the subsystems of second order arithmetic that we shall consider employ classical logic and include the basic arithmetical axioms and the restricted induction axiom

\begin{displaymath}(0\in W\wedge\forall n\,(n\in W\rightarrow n+1\in
W))\rightarrow\forall n\,(n\in W)\,.\end{displaymath}

Two of the most important subsystems of second order arithmetic are $\mathsf{ACA}_0$ and $\Pi^1_1$- $\mathsf{CA}_0$. A formula of the language of second order arithmetic is said to be arithmetical if it contains no set quantifiers. The axioms of $\mathsf{ACA}_0$ consist of the basic arithmetical axioms, the restricted induction axiom, and arithmetical comprehension, i.e., the comprehension scheme for formulas $\varphi(n)$ which are arithmetical. A $\Pi^1_1$ formula is one of the form $\forall
W\,\theta$ where $\theta$ is arithmetical. The axioms of $\Pi^1_1$- $\mathsf{CA}_0$ consist of the axioms of $\mathsf{ACA}_0$ plus $\Pi^1_1$ comprehension. Obviously $\Pi^1_1$- $\mathsf{CA}_0$ is much stronger than $\mathsf{ACA}_0$. Three other very important subsystems of second order arithmetic are $\mathsf{RCA}_0$ and $\mathsf{WKL}_0$, both of which are weaker than $\mathsf{ACA}_0$, and $\mathsf{ATR}_0$, which is intermediate between $\mathsf{ACA}_0$ and $\Pi^1_1$- $\mathsf{CA}_0$. For background material on subsystems of second order arithmetic, we refer the reader to [12,5,6,23].

The purpose of this section is to show how some fundamental results concerning separable Banach spaces and the weak-* topology can be developed formally within $\mathsf{ACA}_0$ and weaker systems. In particular, we show that a version of the Krein-Smulian theorem is provable in $\mathsf{ACA}_0$. Our approach for the development of separable Banach space theory within subsystems of second order arithmetic follows that of Brown and Simpson [5,7,4,6,23]; see also the paper of Shioji and Tanaka [22].

Definition 4.1 (tex2html_wrap_inline$RCA_0$)   A (code for a) complete separable metric space $\widehat A$ is defined to be a set $A\subseteq{\mathbb{N} }$ together with a function $d:A\times A\to{\mathbb{R} }$ such that, for all $a,b,c\in A$,
1.
d(a,a)=0,
2.
d(a,b)=d(b,a), and
3.
$d(a,c)\le d(a,b)+d(b,c)$.
A (code for a) point of $\widehat A$ is defined to be a sequence $\langle{}a_n\mid n\in{\mathbb{N} }\rangle{}$ of elements of A such that $\forall
m\,\forall n\,(m<n\longrightarrow d(a_m,a_n)\le1/{2^m})$. Although $\widehat A$ does not formally exist as a set within $\mathsf{RCA}_0$, we use notations such as $x\in{\widehat A}$ to mean that x is a point of $\widehat A$, etc. We then straightforwardly extend our definitions of = and d to  $\widehat A$ in such a way that $\langle{}\widehat A,d\rangle{}$ is a complete metric space, with A dense in  $\widehat A$.

Definition 4.2 (tex2html_wrap_inline$RCA_0$)   Let $\widehat A$ be a complete separable metric space as defined above. A (code for an) open set in $\widehat A$ is defined to be a sequence of ordered pairs $U=\langle{}(a_i,q_i)\mid i\in{\mathbb{N} }\rangle{}$, $a_i\in A$, $q_i\in{\mathbb{Q} }$. We write $x\in U$ to mean that $x\in{\widehat A}$ and d(x,ai)<qifor some $i\in{\mathbb{N} }$. A closed set in $\widehat A$ is defined to be the complement of an open set.  

Building on the definitions above within $\mathsf{RCA}_0$, one can define corresponding notions of continuous function from one complete separable metric space into another, etc. On this basis, one can prove within $\mathsf{ACA}_0$ or $\mathsf{WKL}_0$ or $\mathsf{RCA}_0$ many fundamental results about the topology of complete separable metric spaces. For details, see [23]. In particular, one can prove within $\mathsf{WKL}_0$ (see Chapter IV of [23]) the Heine-Borel covering lemma (``if Cis compact then any covering of C by a sequence of open sets has a finite subcovering''), using the following $\mathsf{RCA}_0$ notion of compactness:

Definition 4.3 (tex2html_wrap_inline$RCA_0$)   Let C be a closed set in a complete separable metric space $\widehat A$. We say that C is compact if there exists a countable sequence of finite sequences of points $\langle{}\langle{}
x_{ni}\mid i\le k_n\rangle{}\mid n\in{\mathbb{N} }\rangle{}$ in $\widehat A$ such that for all $x\in C$ and all $n\in{\mathbb{N} }$ there exists $i\le k_n$ such that d(x,xni)<1/2n.

In the same setting, there is a useful version of the Tychonoff product theorem, and one can prove within $\mathsf{RCA}_0$ the compactness of the product space $\prod_{n\in{\mathbb{N} }}[a_n,b_n]$, where $\langle{}[a_n,b_n]\mid
n\in{\mathbb{N} }\rangle{}$ is any sequence of closed bounded intervals. For details, see Chapter IV of [23].

We now turn to our development of separable Banach space theory within $\mathsf{RCA}_0$ and $\mathsf{ACA}_0$.

Definition 4.4 (tex2html_wrap_inline$RCA_0$)   A (code for a) separable Banach space consists of a countable set $A\subseteq{\mathbb{N} }$ together with operations $+:A\times A\to A$, $-:A\times A\to A$, and $\cdot:{\mathbb{Q} }\times A\to A$ and a distinguished element $0\in A$ such that $\langle{}A,+,-,\cdot,0\rangle{}$forms a countable vector space over the rational field ${\mathbb{Q} }$, together with a function $\Vert\,\Vert:A\to{\mathbb{R} }$ satisfying
1.
$\Vert qa\Vert=\vert q\vert\Vert a\Vert$ for all $a\in A$ and $q\in{\mathbb{Q} }$, and
2.
$\Vert a+b\Vert\le\Vert a\Vert+\Vert b\Vert$ for all $a,b\in A$.
In other words, a code for a separable Banach space $\widehat A$ is a countable pseudo-normed vector space A over ${\mathbb{Q} }$. Note that $\widehat A$ is a complete separable metric space under $d(a,b)=\Vert a-b\Vert$. Thus a point of the separable Banach space $\widehat A$ is by definition a sequence $\langle{}a_n\mid n\in{\mathbb{N} }\rangle{}$ such that $\forall
m\,\forall n\,(m<n\longrightarrow\Vert a_m-a_n\Vert<1/{2^m})$.

Definition 4.5 (tex2html_wrap_inline$RCA_0$)   Let $X={\widehat A}$ and $Y={\widehat B}$ be separable Banach spaces. A (code for a) bounded linear operator $F:X\to Y$ is a linear mapping $F:A\to{\widehat B}$ such that, for some $0\le r<\infty$, $\Vert F\Vert\le r$, i.e., $\Vert F(a)\Vert\le
r\Vert a\Vert$ for all $a\in A$. If $x=\langle{}a_n\mid n\in{\mathbb{N} }\rangle{}$ is a point of $X={\widehat A}$, we write $F(x)=\lim_nF(a_n)$.

It can be proved in $\mathsf{RCA}_0$ [5,7,23] that bounded linear operators $F:X\to Y$ are identifiable with continuous linear mappings from X into Y. Also within $\mathsf{RCA}_0$ one can prove a useful version of the Banach-Steinhaus theorem:

Theorem 4.6   The following is provable in $\mathsf{RCA}_0$. Given separable Banach spaces Xand Y and a sequence of bounded linear operators $F_n:X\to Y$, $n\in{\mathbb{N} }$, if $\{\Vert F_n(x)\Vert\mid n\in{\mathbb{N} }\}$ is bounded for all $x\in X$, then there exists $r<\infty$ such that $\Vert F_n\Vert\le r$ for all $n\in{\mathbb{N} }$.  

Proof. See [7,6,23].

Definition 4.7 (tex2html_wrap_inline$RCA_0$)   Let X be a separable Banach space. A bounded linear functional on X is a bounded linear operator $f:X\to{\mathbb{R} }$. We write $f\in X^*$ to mean that f is a bounded linear functional on X. For $0\le r<\infty$, we write $f\in B_r(X^*)$ to mean that $f\in X^*$ and $\Vert f\Vert\le r$.

Note that X* and Br(X*) do not formally exist as sets within $\mathsf{RCA}_0$. We identify the functionals in Br(X*) in the obvious way with the points of a certain closed set in the compact metric space $\prod_{a\in A}[-r\Vert a\Vert,r\Vert a\Vert]$, where $X={\widehat A}$. Thus the compactness of Br(X*) is provable in $\mathsf{RCA}_0$. This version of the Banach-Alaoglu theorem turns out to be very useful for the development of separable Banach space theory within $\mathsf{WKL}_0$. See [5,22] and Chapter IV of [23] and Brown's discussion of the ``Alaoglu ball'' [7]. In particular we have:

Theorem 4.8   The following version of the Hahn-Banach theorem is provable in $\mathsf{WKL}_0$. Let X be a separable Banach space and let Y be a subspace of X. If $g:Y\to{\mathbb{R} }$ is a bounded linear functional with $\Vert g\Vert\le r$, then there exists a bounded linear functional $f:X\to{\mathbb{R} }$ such that $\Vert f\Vert\le r$ and f extends g, i.e., f(x)=g(x) for all $x\in Y$.  

Proof. The literature contains two proofs of this result. A direct proof is in [5]. An indirect proof via a $\mathsf{WKL}_0$ version of the Markov-Kakutani fixed point theorem is in [22] and Chapter IV of [23].

Theorem 4.9   The following extension of the Hahn-Banach Theorem is provable in $\mathsf{WKL}_0$. Let X be a separable Banach space. Let $p:X\to{\mathbb{R} }$ be a continuous function such that p(rx)=rp(x) and $p(x+y)\le p(x)+p(y)$for all $r\ge0$ and $x,y\in X$. Let Y be a subspace of X and let $g:Y\to{\mathbb{R} }$ be a bounded linear functional such that $g(x)\le p(x)$ for all $x\in Y$. Then there exists a bounded linear functional $f:X\to{\mathbb{R} }$ such that f extends g and $f(x)\le p(x)$ for all $x\in X$.  

Proof. Either of the cited proofs of Theorem 4.8 can be straightforwardly adapted to prove this more general result. See also Theorem 4.2 of [7].

For use later in this section, we note that the following separation principle holds in $\mathsf{ACA}_0$:

Lemma 4.10   The following is provable in $\mathsf{ACA}_0$. Let X be a separable Banach space. Let Z be a countable set in X such that $\vert\vert x\vert\vert\ge1$ for all x in the convex hull of Z. Then there exists $f\in B_1(X^*)$such that $f(x)\ge1$ for all x in the convex hull of Z.  

Proof. Put

\begin{displaymath}W=\Biggl\{\sum_{i<n}q_iz_i\Bigm\vert
n\in{\mathbb{N} }\,,\;q_...
...athbb{Q} }\cap[0,1]\,,\;\sum_{i<n}q_i=1\,,\;z_i\in Z\Biggr\}\,.\end{displaymath}

Note that $\Vert w\Vert\ge1$ for all $w\in W$. Fix $x_0\in W$. By arithmetical comprehension, there is a continuous function $p:X\to{\mathbb{R} }$defined by

\begin{displaymath}p(x)=\inf\{c\in{\mathbb{Q} }\mid
c>0\wedge\exists w\in W\,(\Vert(x/c)+w-x_0\Vert\le1)\}\,.\end{displaymath}

Then p satisfies the following:
1.
$p(x_0)\ge1$.
2.
$0\le p(x)\le\Vert x\Vert$ for all $x\in X$.
3.
p(rx)=rp(x) for all $r\ge0$ and $x\in X$.
4.
$p(x+y)\le p(x)+p(y)$ for all $x,y\in X$.
Some remarks: if property 1 failed, then there would be $w\in W$ and a rational c<1 such that $\Vert(x_0/c)+w-x_0\Vert\le1$, whence $\Vert(1-c)x_0+cw\Vert\le c<1$, contrary to (1-c)x0+cw being in W. Properties 2 and 3 are easily verified. As for property 4, given c>p(x)+p(y), write c=a+b where a>p(x) and b>p(y). Then there exist w1 and w2 in W such that $\Vert(x/a)+w_1-x_0\Vert\le1$ and $\Vert(y/b)+w_2-x_0\Vert\le1$. Then

\begin{displaymath}\frac{x+y}{a+b}+\frac{a}{a+b}w_1+\frac{b}{a+b}w_2-x_0
=\left(...
...a}+w_1-x_0)+
\left(\frac{b}{a+b}\right)(\frac{y}{b}+w_2-x_0)\,,\end{displaymath}

so

\begin{displaymath}\left\Vert\frac{x+y}{a+b}+\frac{a}{a+b}w_1+\frac{b}{a+b}w_2-x_0\right\Vert
\le \frac{a}{a+b}+\frac{b}{a+b}=1\end{displaymath}

whence $p(x+y)\le a+b=c$.

Let Y be the subspace of X generated by x0, i.e., $Y={\mathbb{R} }x_0$. Define $g:Y\to{\mathbb{R} }$ by g(rx0)=rp(x0). Then g is a bounded linear functional on Y. Moreover, if $r\ge0$ then g(rx0)=rp(x0)=p(rx0), and if r<0 then $g(rx_0)=rp(x_0)\le0\le p(rx_0)$, so $g\le p$ on Y. Thus, by our extended Hahn-Banach theorem 4.9 in $\mathsf{WKL}_0$, we can extend g to a bounded linear functional $f:X\to{\mathbb{R} }$ such that $f\le p$ on X.

Let $w\in W$ be given, and suppose y is such that $\Vert y\Vert\le1$. Then $\Vert(y-w+x_0)+w-x_0\Vert=\Vert y\Vert\le1$, whence $f(y-w+x_0)\le p(y-w+x_0)\le1$by definition of p. But f(y-w+x0)=f(y)-f(w)+f(x0) and $f(x_0)\ge1$, so $f(y)\le f(w)$. Replacing f by $f/\Vert f\Vert$, we see that $f\in B_1(X^*)$ and $f(x)\ge1$ for all x in the convex hull of Z. This completes the proof.

We conjecture that this separation principle is actually provable in $\mathsf{WKL}_0$ and not only in $\mathsf{ACA}_0$.

We now begin our treatment of the weak-* topology within $\mathsf{RCA}_0$. We start by introducing an $\mathsf{RCA}_0$ version of the bounded-weak-*topology:

Definition 4.11 (tex2html_wrap_inline$RCA_0$)   A (code for a) bounded-weak-*-closed set C in X* is defined to be a sequence of (codes for) closed sets $C_n\subseteq
B_n(X^*)$, $n\in{\mathbb{N} }$, such that

\begin{displaymath}\forall m\,\forall
n\,(m<n\longrightarrow C_m=B_m(X^*)\cap C_n)\,.\end{displaymath}

We write $x^*\in C$to mean $\exists n\,(x^*\in C_n)$, or equivalently $\forall
n(n>\Vert x^*\Vert\rightarrow x^*\in C_n)$. A bounded-weak-*-open set in X* is defined to be the complement of a bounded-weak-*-closed set in X*.  

The next lemma formalizes within $\mathsf{ACA}_0$ a well-known fact (see Lemma V.5.4 in [11]): there is a bounded-weak-*neighborhood basis of 0 in X* consisting of the polars of sequences converging to 0 in X.

Lemma 4.12   The following is provable in $\mathsf{ACA}_0$. Let X be a separable Banach space. If $\langle{}x_n\mid n\in{\mathbb{N} }\rangle{}$ is a sequence of points in X such that $x_n\to0$, then

\begin{displaymath}\{x^*\in X^*\mid\forall n\,\vert x^*(x_n)\vert<1\}\end{displaymath}

contains 0 and is bounded-weak-*-open in X*. Conversely, if U is a bounded-weak-*-open set in X* containing 0, then we can find a sequence of points $\langle{}x_n\mid n\in{\mathbb{N} }\rangle{}$ in X such that $x_n\to0$ and $\{x^*\in X^*\mid\forall n\,\vert x^*(x_n)\vert\le1\}\subseteq
U$.  

Proof. Reasoning in $\mathsf{ACA}_0$, let $\langle{}x_n\mid n\in{\mathbb{N} }\rangle{}$ be a sequence of points in X such that $\lim_nx_n=0$. By arithmetical comprehension, there exists a sequence of integers Nm, $m\in{\mathbb{N} }$, such that $\Vert x_n\Vert<1/m$for all $m\in{\mathbb{N} }$ and $n\ge N_m$. Using the sequence $\langle{}N_m\mid
m\in{\mathbb{N} }\rangle{}$ as a parameter, we can define a sequence of closed sets $C_m\subseteq B_m(X^*)$, $m\in{\mathbb{N} }$, by

\begin{displaymath}C_m=\{x^*\in
B_m(X^*)\mid\exists n<N_m(\vert x^*(x_n)\vert\ge1)\}\,.\end{displaymath}

It is easy to verify that

\begin{displaymath}C_m=\{x^*\in B_m(X^*)\mid\exists n\,\vert x^*(x_n)\vert\ge1\}\end{displaymath}

and hence $C_k=C_m\cap B_k(X^*)$ for all k<m. Thus by Definition 4.11 we have a bounded-weak-*-closed set $C=\bigcup_{m\in{\mathbb{N} }}C_m$ and clearly

\begin{displaymath}X^*\setminus C=\{x^*\in
X^*\mid\forall n\,\vert x^*(x_n)\vert<1\}\,.\end{displaymath}

This shows that $\{x^*\in
X^*\mid\forall n\,\vert x^*(x_n)\vert<1\}$ is bounded-weak-*-open in X*.

The proof of the converse will be carried out in $\mathsf{WKL}_0$. Let U be a bounded-weak-*-open set in X* containing 0. Then $C=X^*\setminus U$ is a bounded-weak-*-closed set with $0\notin C$. For any countable set $S\subseteq X$ let So be the polar of S, i.e.,

\begin{displaymath}S^o=\left\{x^*\in X^*\bigm\vert\forall x\in S\,\vert x^*(x)\vert\le
1\right\}\,.\end{displaymath}

To complete the proof, we need to construct a sequence of points $\langle{}x_k\mid k\in{\mathbb{N} }\rangle{}$ in X such that $\lim_kx_k=0$ and $\{x_k\mid k\in{\mathbb{N} }\}^o\cap C=\emptyset$.

Let $X={\widehat A}$ where A is a countable dense set in X. Put A0=A and, for each $n\ge1$, $A_n=\{a\in A\mid\Vert a\Vert<1/n\}$. Claim: for any $n\in{\mathbb{N} }$ and any countable set $S\subseteq X$, if $S^o\cap B_n(X^*)\cap C=\emptyset$ then there exists a finite set $F\subset A_n$ such that $(S\cup F)^o\cap B_{n+1}(X^*)\cap
C=\emptyset$. If such an F does not exist, then for all finite sets $F\subset A_n$ we would have $(S\cup F)^o\cap B_{n+1}(X^*)\cap
C\neq\emptyset$, so by the Heine-Borel covering property of the compact set Bn+1(X*) it would follow that

\begin{displaymath}(S\cup
A_n)^o\cap B_{n+1}(X^*)\cap C\neq\emptyset\,.\end{displaymath}

But $(S\cup
A_n)^o=S^o\cap A_n^o=S^o\cap B_n(X^*)$, and hence $S^o\cap B_n(X^*)\cap C\neq\emptyset$, a contradiction. This proves the claim.

Within $\mathsf{WKL}_0$, we can apply the claim above repeatedly starting with $F_0=\emptyset$ to obtain a sequence of finite sets $F_{n+1}\subseteq
A_n$, $n\in{\mathbb{N} }$, such that

\begin{displaymath}(F_0\cup\ldots\cup F_n)^o\cap
B_n(X^*)\cap C=\emptyset\end{displaymath}

for all $n\in{\mathbb{N} }$. The construction can be carried out effectively within $\mathsf{WKL}_0$ because, by Lemma 5.8 of [3], the predicates $F\subset A_n$ and $F^o\cap B_n(X^*)\cap C=\emptyset$ are provably in $\mathsf{WKL}_0$ equivalent to $\Sigma^0_1$ formulas. Thus the existence of a sequence of finite sets $\langle{}F_n\mid n\in{\mathbb{N} }\rangle{}$ with the mentioned properties is provable in $\mathsf{WKL}_0$.

Letting $\langle{}x_k\mid k\in{\mathbb{N} }\rangle{}$ be an enumeration without repetition of $\bigcup_{n\in{\mathbb{N} }}F_n$, it is clear that $x_k\to0$ in X and that $\{x_k\mid k\in{\mathbb{N} }\}^o\cap C=\emptyset$. This completes the proof.

We now introduce our $\mathsf{RCA}_0$ version of the weak-* topology.

Definition 4.13 (tex2html_wrap_inline$RCA_0$)   A weak-*-open set in X* is defined to be a bounded-weak-*-open set U in X* such that for all $x_0^*\in U$there exists a finite sequence of points $x_0,\ldots,x_{n-1}\in X$such that

\begin{displaymath}\{x^*\in X^*\mid\forall
k<n\,(\vert x^*(x_k)-x_0^*(x_k)\vert\le1)\}\subseteq U\,.\end{displaymath}

A weak-*-closed set in X* is defined to be the complement of a weak-*-open set in X*.  

According to the previous definition, we have trivially in $\mathsf{RCA}_0$ that any weak-*-closed set is bounded-weak-*-closed. In $\mathsf{ACA}_0$ we have following version of the Krein-Smulian theorem:

Theorem 4.14   The following is provable in $\mathsf{ACA}_0$. Let X be a separable Banach space. Suppose that $C\subseteq X^*$ is convex and bounded-weak-*-closed. Then C is weak-*-closed.  

Proof. Let $x_0^*\notin C$ be given. Then C-x0* is also bounded-weak-*-closed and convex, and $0\notin C-x_0^*$. Thus $(C-x_0^*)\cap
B_n(X^*)$ is a closed subset of Bn(X*) for each $n\in{\mathbb{N} }$. Since Bn(X*) is compact, in $\mathsf{ACA}_0$ there exists a countable dense subset $D_n\subset (C-x_0^*)\cap B_n(X^*)$ for each n (see Theorem 3.2 in [4]). We want to find a weak-*-open set N containing 0 in X* which is disjoint from C-x0*, as this will show that x0*+N is disjoint from C. Let U be a bounded-weak-*-open set containing 0 which is disjoint from C-x0*. By Lemma 4.12, there is a sequence $\{x_n\mid n\in{\mathbb{N} }\}$ such that $x_n\to0$ and $\{x^*\in X^*\mid\forall
n\vert x^*(x_n)\vert\le 1\}\subseteq U$.

Now, for each $m\in{\mathbb{N} }$, define a function $T_m:B_m(X^*)\to c_0$ by $T_m(x^*)=\langle{}x^*(x_n)\mid n\in{\mathbb{N} }\rangle{}$. These form a sequence of compatible functions, i.e., if m<n then Tn|Bm(X*)=Tm. Thus we can define a function $T:X^*\to c_0$ by T(x*)=Tm(x*) where $x^*\in B_m(X^*)$. Notice that T is linear, and $\Vert T(x^*)\Vert\le1$implies $x^*\in U$ for all $x^*\in X^*$. Let D be an enumeration of the dense sets $D_n\subset (C-x_0^*)\cap B_n(X^*)$, and let E=T(D)(which exists by arithmetical comprehension); then E is a countable subset of c0, and $\Vert x\Vert>1$ for all x in the convex hull of E. By Lemma 4.10 there exists $f\in B_1(c_0^*)=B_1(\ell_1)$ such that $f(x)\ge1$ for all x in the convex hull of E. Write $f=\langle{}\alpha_n:n\in{\mathbb{N} }\rangle{}\in\ell_1$.

Let $x=\sum_{n\in{\mathbb{N} }}\alpha_nx_n$; note that $\{x^*\in X^*\mid
\vert x^*(x)\vert<1\}$ is weak-*-open and contains 0. Also, if $y^*\in C-x_0^*$ then $y^*\in B_m(X^*)$ for some m. Thus $\vert y^*(x)\vert=\vert\sum_{n\in{\mathbb{N} }}\alpha_ny^*(x_n)\vert=\vert f(T(y^*))\vert=\vert f(T_m(y^*))\vert\ge1$since Dm is dense in Bm(X*) and f and Tm are continuous. So let $N=\{x^*\in X^*\mid \vert x^*(2x)\vert<1\}=\{x^*\in X^*\mid
\vert x^*(x)\vert<1/2\}$; then N is weak-*-open, contains 0, and is disjoint from C-x0*. This completes the proof.

Specializing to subspaces of X* (see also Corollary 2.8 above), we obtain:

Corollary 4.15   The following is provable in $\mathsf{ACA}_0$. Let X be a separable Banach space. Let C be a closed set in B1(X*) such that $C=B_1(X^*)\cap\mbox{\rm span}(C)$ where

\begin{displaymath}\mbox{\rm span}(C)=\left\{\sum_{i=0}^na_i\,x^*_i\Bigm\vert a_i\in{\mathbb{R} }\,,\,x^*_i\in
C\,,\,n\in{\mathbb{N} }\right\}\,.\end{displaymath}

Then $\mbox{\rm span}(C)$ is a weak-*-closed subspace of X*.  

Proof. This follows easily from the previous theorem. See also the proof of Corollary 2.8.


next up previous
Next: Proof of the main Up: Separable Banach space theory Previous: Trees and subspaces of
Stephen G Simpson
1998-10-25