\documentstyle[tr_def,amssymbo,twoside,paper]{article}
\soyad{™NAL}
\author{S�leyman ™nal}
\title{COMPLEMENTED SUBGROUPS OF PRODUCT OF TOPOLOGICAL GROUPS\footnote
{Mathematics Subject Classification Primary 54H11, 46A05. Secondary 22AO5 
Key words and phrasses. 
Complemented subgroups, product of topological groups.}
}
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\begin{document}
\maketitle
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\begin{abstract}
It is proved that every complemented subgroup of a product of topological
groups is homeomorphic to a product of topological groups which are complemented
in subproducts with cardinality at most ${\em m}$, if the groups do not contain a copy of 
Cantor cube with weight ${\em m}^+$. 

An application of this result yields to the fact 
that complemented subspaces of a product of 
Frechet spaces are product of Frechet spaces, 
themselves. In particular, the problem of 
characterization of complemented subspaces of arbitrary products of Banach spaces is
reduced to countable products, and furthermore this characterizes injective objects
in the category of locally convex spaces as product of injective 
Frechet spaces. 
\end{abstract}

\baslik{Introduction}
In $[2]$, Domanski and Ortynski showed that complemented subspaces of powers
of $l_p$ and $c_0$ spaces are of the same kind, respectively. In $[1]$, 
Domanski also showed that, complemented subspaces of product of 
Hilbert spaces, themselves, 
are homeomorphic to product of Hilbert spaces. Those are partial answers, 
depending on special properties of the mentioned 
spaces and using Pelczynski decomposition
method and transfinite induction, to the question whether complemented subspaces of 
products of Banach spaces are of the same 
kind or not $[2]$.

In this article, the author shows that the 
problem of characterization of complemented 
subspaces of product of spaces 
can be reduced to small products. Furthermore, 
since linearity is not required in the method 
used here which includes two main steps, easier and more
direct than method used in [1,2],
the spaces under consideration are extended to topological groups. The 
factorization of functions defined on product spaces $[4]$ 
enables us to obtain lower triangular
matrix representation of the projection under question, 
which is followed by the second step of showing for the projection having
the above property, the restrictions of diagonals to the complemented subgroup, 
say $H$, in the said represantation are isomorphism between 
$H$ and product of image of diagonal elements. 

The following notation and terminology is used. $+, 0$ denote group operation and 
unity, respectively. Let $I$ be an index set and $A \subset I$, 
we identify
$G_A=\db \prod_{i \in A} G_i$ as a subgroup of $G=\db \prod_{i \in I} G_i$ and
$\pi_A: G \ra G_A$ as 
natural projections. \\
$T: \db \prod_{i \in J}  G_i \ra \db \prod_{j  \in I} H_j$, 
a group homomorphism and we define
$T_{BA}: G_A \ra H_B$ natural map induced by $T$, i.e 
$\pi_B T$, 
where $A \subset I$ and $ B \subset J. \quad (T_{BA})_{(B,A) \in Q \times P}$
denotes the matrix of $T$ with respect 
to partitions $Q$ and $P$ of $J$ and $I$,
respectively, and we use the term lower triangular, 
if $I=J$, $P=Q$ and $P$ is ordered, in the 
usual meaning. Following the standard
terminology, an ordinal is considered 
as a set of smaller ordinals and a cardinal is 
an initial ordinal, i.e. not equipotent 
to smaller ordinals. For a set $S \; card (S)$ denotes
the cardinality of $S. \;\; \Sigma_{< {\em l}}$ 
product of $G_i, i \in I$ is the 
subgrouup of $\db \prod_{i \in I} G_i$, which consists of elements with the number of 
coordinates different from $O$, is less than 
${\em k}. <{\em l}$ box topology is 
topology defined on $\db \prod_{i \in I} G_i$, generated by open boxes 
$\prod_{i \in I} V_i$ with the set of $V_i$ which are not equal to $G_i$ has 
has cardinality less than ${\em l}$.

In the first part of the 
article, topological groups are
considered, and then the results
are applied to the Banach and Frechet spaces. 
Specifically,  it is shown that complemented 
subspaces of product of Frechet spaces are 
product of Frechet spaces. 

The method also can be applied to $\sum_{<{\em k}}$
products and box topologies. 

\noindent {\bf 1. Projections which have lower triangular matrices}

Following lemmas are purely algebraic, 
first one is just matrix multiplication and  the 
second one is an easy consequence of  the first. 

\begin{claim}{Lemma 1.1:} Let $P: G_1 \times G_2 \ra G_1 \times G_2$ be a
group homomorphism where
$G_1$ and $G_2$ are arbitrary groups. 

If
$P^2=P$ and
$P=\left(
\begin{array}{cc}
P_{\{1\}\{1\}} & 0\\
P_{\{2\}\{1\}} & P_{\{2\}\{2\}}
\end{array}
\right)
$
then $P^2_{\{1\}\{1\}}=P_{\{1\}\{1\}},$\\
$P^2_{\{2\}\{2\}}=
P_{\{2\}\{2\}}$ and 
$P_{\{2\}\{2\}} P_{\{2\}\{1\}} 
P_{\{1\}\{1\}}=0$. 
\end{claim}

\begin{claim}{Lemma 1.2:}
Let $G$ be a subgroup of the product of a family of groups
$\{G_i |i \in {\em m}\}$ where 
${\em m}$ is a cardinal. 
If $P: G \ra G$ is an idempotent group
homomorphism and \\
$P=(P_{\{i\}\{j\}})_{(i,j) \in {\em m} \times {\em m}}$ 
is lower triangular 
and $P_{\{i\} (j,m)}=0$
for each $i \leq j<m$ then
for each order convex subset of $A$ of ${\em m}$ 
and for each ordered partition $A_1, A_2$ 
of $A$ we have $P^2_{AA}=P_{AA}$ and 
$P_{A_2 A_2} P_{A_2A_1} P_{A_1A_1}=0$. In particular 
$P^2_{\{i\} \{i\}}=P_{\{i\} \{i\}}$ and 
$P_{\{i\} \{i\}} P_{\{i\} i} P_{ii}=0$ for each ordinal 
$i \in {\em m}$. Here $i=\{j|j<i\}$. 
\end{claim}

Following is the main result of this article, although it seems
to be rather technical. Other results are consequences 
of this result and the factorization of a 
map on product spaces over the small product. 
Proof of this fact only requires Lemma 1.2
and transfinite induction. 

\begin{claim}{Theorem 1.3:}
Let $H$ be a complemented subgroup of the product of topological groups
$\{G_i |i \in m\}$ with continuous projection $P$. 
If $P=(P_{\{i\} \{j\}})_{(i,j) \in {\em m} \times m}$
is lower triangular then
$H \cong  \db \prod_{i \in {\em m}} P_{\{i\} \{i\}}(G_i)$ 
and also the restriction of the diagonal
of $P$ to $H$ is an homeomorphism and 
$P_{\{i\} \{i\}}(G_i)$ is complemented in $G_i$ for each $i \in {\em m}$.
\end{claim}

\begin{claim}{Proof:}
Let $Q=(\delta_{ij} P_{\{i\} \{i\}})_{(i,j) \in {\em m} \times {\em m}}$
be the diagonal of $P$ and $S$ be the restriction of $Q$
to $H$. First we will define $T: \db \prod_{i \in {\em m}} 
P_{\{i\} \{i\}}(G_i) \ra
\db \prod_{i \in {\em m}} G_i$ by transfinite recursion and then we 
will show that $T$ is a continuous group homemorpism with the  range
in $H$ and $ST=I, TS=I$, by 
induction. Let $(y_i)_{i \in {\em m}} \in \db \prod_{i \in {\em m}} 
P_{\{i\} \{i\}}(G_i)$ and 
define by recursion $x_i \in G_i$ as follows:
$x_0=y_0$ and $x_i= P_{\{i\} i} (x_j)_{j<i} +y_i$. 
Set $T(y_i)_{i<m}=(x_i)_{i<m}$. 
It is obvious that $T$ is a group homemorphism. 
To show that $T$ is continuous,
it is sufficient to show that 
$\pi_{\{i\}} T$ is continuous for
each 
$i \in {\em m}. \;\; \pi_{\{0\}} T$ is already continuous
by definition of $T$. Now suppose $\pi_{\{j\}} 
T$ for each $j<i$
for some 
$0 \neq i \in m$. 
Then $\pi_i T$ is continuous and 
$\pi_{\{i\}} T= P_{\{i\}i} \pi_i T+ \pi_{\{i\}}$
are both continuous. Hence induction gives us continuity of $T$. Next
we have to show $T(\db \prod_{i \in {\em m}} P_{\{i\} \{i\}}(G_i)) \subset H$. 
Let $(x_i)_{i \in {\em m}}=T(y_i)_{i \in {\em m}}$. 
We will show that $P(x_i)_{i \in m}=(x_i)_{i \in {\em m}}$
and then 
$P_{\{i\} \{i\}}(x_i)=y_i$
\end{claim}

\noindent We have $x_0=y_0,y_0 \in P_{\{0\} \{0\}}(G_0)$
and $\pi_{\{0\}}P(x_i)_{i \in {\em m}}= P_{\{0\} \{0\}} x_0=x_0$ 
by lemma 1.2. \\
Suppose
$\pi_j P(x_i)_{i \in m}=(x_i)_{i \in j}$ some $0 \neq j \in m$. 
Then we have\\
$\star \quad \pi_{\{j\}}  P(x_i)_{i \in {\em m}}=
P_{\{j\}j} (x_i)_{i \in j}+
P_{\{j\} \{j\}} x_j+
P_{\{j\} (j,m)} (x_i)_{j<i<{\em m}}$


But the last term is zero by the lower triangularity of the matrix
of $P$. We have $\pi_j P(x_i)_{i \in m}=P_{jj} (x_i)_{i \in j}$ 
by the lower
triangulity and $\pi_j P(x_i)_{i \in m}=(x_i)_{i \in j}$ by the 
choice of $j$. So we get $(x_i)_{i \in j}=P_{jj} (x_i)_{i \in j}$ 
and $P_{\{j\} \{j\}} 
P_{\{j\} j} (x_i)_{i \in j}=0$ by lemma 1.2. Hence 
\begin{eqnarray*}
P_{ \{j\} \{j\} } (x_j) &=& P_{ \{j\} \{j\} } 
(P_{\{j\}j} (x_i)_{i < j} + x_j)\\
&=& P_{\{j\} \{j\}} (y_j)\\
&=& (y_j)
\end{eqnarray*} 
by the definition of $(x_j)$ and $y_j \in 
P_{\{j\} \{j\}} G_j$. So
$\pi_{\{j\}} P(x_i)_{i \in m}=y_j$ by $(\star)$.
So the induction step is completed. 
Hence $P(x_i)_{i \in m}=(x_i)_{i \in m}$. 
Now we have 
\begin{eqnarray*}
P_{\{i\} \{i\}}(x_i) &=& P_{\{i\} \{i\}} 
P_{\{i\} i} (x_j)_{j \in i}+
P_{\{i\} \{i\}} (y_i)\\
&=& 0+ P_{\{i\} \{i\}} (y_i)\\
&=& y_i
\end{eqnarray*}
Hence $S(x_i)_{i \in m}=(P_{ii}(x_i))_{i \in m}=(y_i)_{i \in m}$

Therefore $S T$ is the identity of $\db \prod_{i \in {\em m}} 
P_{\{i\} \{i\}}
(G_i)$. 
It remains to be shown that $T S$ is the identity of $H$ and to do so 
it is sufficient to show that $S$ is an injection. 
Suppose $S(x_i)_{i \in {\em m}}=0$
and $(x_i)_{i \in {\em m}} \in H$. After an
easy calculation we get $x_0=0$ and $x_i=0$ when $(x_j)_{j \in i}=0$, 
since $x_i=P_{\{i\} i} (x_j)_{j \in i}+P_{\{i\} \{i\}}(x_i)$ and
$\pi_{\{i\}} S(x_k)_{k \in m}=
P_{\{i\} \{i\}} (x_i)=0$. This completes the proof. 
\mb\vs{0.5cm}

In the above theorem, 
$H$ can be  taken as complemented subgroup of 
$\sum_{< {\em k}}$ product when ${\em k}$ is a 
regular cardinal. In that case $H$ is homeomorphic
to the $\sum_{< {\em k}}$ product of the 
family $\{P_{\{i\} \{i\}}G_i|i \in {\em m}\}$ in 
canonical way as in the above theorem. 
Proof of this can be done in same way but the 
only difficulty occurs when defining
$T$,  one must guarentee that $(x_j)_{j \in i}$ are 
already in the $\sum_{<{\em k}}$ product and the 
resultant $(x_i)_{i \in {\em m}}$ also lies in the 
$\sum_{<{\em k}}$ product. 
Other alternate form of above theorem is considering $<- {\em l}$
box topology on $G$
instead of product topology. 
But in this case the continuity 
of the function $T$ 
which is defined in the proof of above theorem can not be guarented
unless assuming each of the group has 
$P_{<{\em l}}$ property. By property $P_{<{\em l}}$ 
we mean that
intersection of any family of open sets which has cardinality less
than ${\em l}$ is still open. 
Of course this property inherits subspace and stable  with
respect to product under $<{\em l}$- box topology when ${\em l}$
is a regular cardinal. So we state theorem 1.3 in stronger form without proof. 

\begin{claim}{Theorem 1.4:} Let $H$ be complemented subgroup of 
$\sum_{< {\em k}}$ product of the family
of topological groups
$\{G_i| i \in {\em m}\}$ under $< l$ box topology where each 
$G_i$ has $P_{<l}$ property 
and $ l,k $ are regular cardinals. 
If $P$ is continuous projection on $H$ 
which has lower triangular matrix 
and $P_{\{i\} (j,m)}=0$
for each $i \leq j<m$ then
then $H$ is homeomorphic to
$\sum_{<{\em k}}$ product of 
$\{P_{\{i\} \{i\}} (G)| i \in m\}$ under 
$< l$ box topology in canonical way. 
\end{claim}

In theorem 1.4 if we take 
${\em l}=\aleph_0$ and ${\em m}{<{\em k}}$ then we get theorem 1.3. 
Theorem 1.4 can also be used in a purely 
algebraic way by taking discrete topology 
on the product or $\sum_{<{\em k}}$ product. 

\newpage
\noindent {\bf 2. Obtaining Lower triangular matrix}

There are several results about continous functions which are defined on product of 
topological spaces. 
One of them states that functions can be factored through small
product under certain assumptions on the spaces
$[3,4]$. In this section 
by using one of these results, namely  
$[4]$ we will get a lower triangular matrix representation
of the projection over small products. 

Following due to Gerlizt $[4]$. 
Here we state the following for the continuous group 
homemorphism which are defined on the product of topological groups. 

\begin{claim}{Lemma 2.1:} 
Let $T: \db 
\prod_{i \in I} G_i \ra H$ be a continuous group homemorphism and 
suppose $H$ 
does not contain a topological copy of the cantor cube 
$\{0,1\}^{{\em m}^+}$. Then there is a set $J$ with 
cardinality at most ${\em m}$, such that 
$J \subset I$  and 
$S: \db \prod_{i \in J} G_i \ra H$ such that $S \pi_J=T$. 
\end{claim}

Note that in the above lemma the condition on $H$, not containing  
a topological copy of $\{0,1\}^{{\em m}^+}$, 
may be replaced with the condition not containing a subgroup of
which is a product ${\em m}^+$ nontrivial 
cylic subgroups and also this condition can be replaced
by $H$ not containing a linear topological 
subspace which is homemorphic to
$m^+$ th power of the field when groups are locally convex spaces and $T$
is linear map. 

Let $T: \db \prod_{i \in I} G_i \ra H$. 
We write  $dep (T) \leq {\em m}$ if $T$
satisfies the conclusion of lemma 2.1. Here we have two lemmas. In the second lemma
we will show that a group homeomorphism $T$ has lower triangular matrix
with respect to a partition  of the 
index set $I$ of the family of topological groups 
$\{G_i |i \in I\}$, 
whoose elements have at most cardinality  
${\em m}$.

\begin{claim}{Lemma 2.2:}
Let $G$ be a subgroup of the product group $\db \prod_{i \in I} G_i$ and 
$T: G \ra G$ be a group homeomorphism. If $dep (\pi_{\{i\}} T) \leq m$ for
each $i \in I$
then for each subset $A$ of $I$ with cardinality at most ${\em m}$, 
there is a subset $B$ of $I$ satisfying 
$A \subset B \;\; card (B) \leq {\em m}$ and
$\pi_B Tx=0$ when $\pi_B (x)=0$. 
\end{claim}

\begin{claim}{Proof:}
Define inductively a sequence of subsets of $I$ with following properties;\\
$A_0=A, \pi_{A_n} Tx=0$ when $\pi_{A_{n+1}} (x)=0$ and
$card (A_n) \leq {\em m}$. 
Suppose that $A_n$ already defined. 
Since $dep (\pi_{\{i\}} \circ T) \leq {\em m}$ for each $i \in I$,
choose $L_i \subset I$ satisfying  card $(L_i) \leq m$ and
$\pi_i (Tx)=0$ when $\pi_{L_i} (x)=0$. Let
$A_{n+1}=\cup \{L_i | i \in A_n\} \cup A_n$.
Hence the induction step is completed. 
Now take $B=\db \cup_{n \in w_0} A_n$ 
which is the desired one. 
\end{claim}

\begin{claim}{Lemma 2.3:}
Let $G$ be subgroup of the product group $\db \prod_{i \in I} G_i$ and 
$T: G \ra G$ be a group homeomorphism. 
If $dep (\pi_{\{i\}} T) \leq {\em m}$ 
then there is a disjoint partition 
$\{J_i| i \in \beta\}$ of $I$ such that 
card $(J_i) \leq m$ for each $i \in \beta$
and the matrix of $T$ with respect to this 
partition is lower triangular i.e. $T_{J_i J_j}=0$ when
$i<j<\beta$ where $\beta$ is an ordinal.
\end{claim}

\begin{claim}{Proof:}
Suppose for an ordinal 
$\delta \;\;(J_i)_{i \in \delta}$ is defined with following properties:

a) $1 \leq card (J_i) \leq {\em m}$. 

b) $J_i \cap J_j=\phi$ when $i,j \in \delta$ and $i \neq j$.

c) $\pi_{J_i} Tx=0$ when 
$\pi_{\cup_{k \leq i} J_{\em k}} (x)=0$
for each $i \in \delta$
\end{claim}

If $I=\cup \{J_i| i \in \delta\}$
then $(J_i)_{i \in \delta}$ is a desired partition otherwise take 
$A$ such that
$\phi \neq A \subset I \setminus \cup \{J_i|i \in \delta\}$ and card $(A)={\em m}$. 

Let $B$ be as in lemma 2.2 for $A$. Then set
$J_\delta=B \setminus \db \cup_{i \in \delta} J_i$ 
The sequence $\{J_i|i \in \delta+1\}$ satisfies the conditions
$a,b,c$. 
This process will finished when
$\{J_i| i \in \beta\}$ cover the set I for some ordinal $\beta$. 
From construction $T_{J_i J_j}=0$ when $i<j<\beta$ where
$T_{J_i J_j}=\pi_{J_i} T \pi_{J_j}$. 

\begin{claim}{Theorem 2.4:}
Let $P: G \ra G$  be continuous projection 
onto complemented subgroup $H$ of $G$ where
$G$ is the product of a family 
$\{G_i| i \in I\}$ of topological groups. If\\ 
\noindent dep 
$(\pi_{\{i\}} P) \leq m$ for each $i \in I$ then there is a partition 
$\{L_\alpha|\alpha \in \delta \}$ of $I$ and complemented subgroup $H_\alpha$ 
of $\db \prod_{i \in L_\alpha} G_i$ with complement $K_\alpha$ such that 
$H \simeq \prod \{H_\alpha|\alpha  \in \delta \}$ and,\\
\noindent kernel $(P)\simeq 
\prod\{K_\alpha|\alpha< \in \delta \}$ and $card(L_\alpha) \leq m$
for each $\alpha \in \delta$.
\end{claim}

\begin{claim}{Proof}: 
By lemma 2.3 we get a partition $\{L_\alpha| \alpha \in \delta \}$ 
of $I$ such that $(P_{L_\alpha L_\beta})_{\alpha,\beta < \delta}$ is lower
tringular. Now result follows theorem 1.3.
\end{claim}

Since space which has a pseudocharacter at most 
$m$ does not contain a copy $\{0,1\}^{m^+}$,
next corollories immediate consequence of above theorem. 

\begin{claim}{Corollary 2.5:}
If $H$ is complemented subgroup of product of family of topological groups
which have psuedocharacter (respectively character) 
at most $m$ then $H$ is also 
product of topological groups which have 
pseudocharacter. (respectively character)
at most $m$. 
\end{claim}

\begin{claim}{Corollary 2.6:}
If $H$ is complemented subgroup of product of metrizable
topological groups then $H$ is also isomorphic to product of metrizable topological groups.
\end{claim}

Next we consider isomorphism between products of topological groups. 

\begin{claim}{Proposition:}
Let $T: \db \prod_{i \in I} G_i \ra \db \prod_{j \in J} H_j$ be 
isomorphism between product of topological
groups
$\db \prod_{i \in I} G_i$ and
$\db \prod_{i \in J} H_j$. 
If $dep (\pi_{\{j\}} T) \leq {\em m}$ and 
$dep(\pi_{\{i\}}  T^{-1}) {\em \leq m}$ 
for each $i \in I$ and 
$j \in J$ then
$card  (I) = card (J)$ and there are partitions
$\{I_\alpha|\alpha<{\em n}\}, 
\{J_\beta|\beta<{\em n}\}$ of 
$I$ and $J$ respectively
such that
$\db \prod_{i \in I_\alpha} G_i \simeq \db \prod_{j \in J_\alpha} H_j$
for each $\alpha< {\em n}$ and
$card(I_\alpha) \leq {\em m}\\
\noindent card (J_\beta) \leq m$ for each 
$\alpha,\beta< {\em n}$ where
${\em n}=card(I)$. 
\end{claim}

\begin{claim}{Proof:}
Using the facts $dep(\pi_{\{j\}} T) \leq {\em m}$ and
$dep ( \pi_{\{j\}} T^{-1}) \leq {\em m}$ 
alternatinaly for a given subset $A$ of I
which has cardinality at most $m$, one can find
subsets $B$ of $I$ and $C$ of $J$ which have 
cordinality at most ${\em m}$ satisfying 
$A \subset B$ and $\pi_c T(x)=0$ 
when $\pi_{B}(x)=0$ 
and 
$\pi_B T^{-1} (x)=0$ when
$\pi_{J} (x)=0$ for any $x$ in the domain of considered 
functions. Continuing in this way and
using tranfinite induction we get $\{I_\alpha|\alpha < {\em n}\}, 
\{J_\beta|\beta <n \}$
partitions of $I$ and $J$ respectively such that, 
$card (I_\alpha) \leq {\em m}\quad card (J_\beta) leq {\em m}$, 
to each $\alpha, \beta \in  {\em n}$ and 
$(T_{I_\alpha J_\beta})_{\alpha,\beta \in {\em n}}$
and $(T^{-1}_{J_\beta I_\alpha})_{\alpha,\beta < {\em n}}$ are lower triangular. Now
it is easy see that $T_{I_\alpha J_\alpha}$ is an isomorphism between 
$\db \prod_{i \in I_\alpha} G_i$ and $\db \prod_{j \in I_\alpha} H_j$. 
\end{claim}

The conclusion in this section can be stated for 
$\sum_{<{\em k}}$ product and
$<{\em l}-$ box topologies by regarding conditions 
of $theorem 1.4.$ 

\noindent {\bf 3. Applications of 
product of Banach and Frechet spaces. }

Now we apply corollary 2.6 to the complemented subspace of product of Frechet spaces
and product of Banach space. This yields following corollaries. 

\begin{claim}{Corollary 3.1:}
If $E$ is a complemented subspace of 
product of Frechet spaces then $E$ is also isomorphic
to product of Frechet spaces. 
\end{claim}

Next corollary characterize injective object in the category 
of locally convex
spaces. 

\begin{claim}{Corollary 3.2}: 
Every injective locally convex space in the category 
of locally convex space
is isomorphic to a product of injective Frechet spaces.
\end{claim}

\begin{claim}{Proof:}
Let $E$ be injective space. Imbed $E$ in $a$ suitable 
$l_\infty(\Lambda)^{\em m}$. Since $E$  is
injective,
$E$ is complemented in $l_\infty (\Lambda)^{\em m}$. By 
$theorem 2.4$, we have
$E \simeq \db \prod_{i \in {\em m}} E_i$ is where each $E_i$ is 
complemented in $l_\infty(\Lambda)^{\aleph_0}$. Hence each $E_i$ is 
injective 
Frechet space and $E$ is isomorphic to the product of these spaces. 
\end{claim}

We use the following lemma to generalize the result of [2], which 
can be proved by using Pelczynski decomposition method. 

\begin{claim}{Lemma 3.3:}
Let $E$ be the countable projective limit of the space $l_p(\Gamma_i)$, where
$\Gamma_i$ is any set which is also complemented in a product of Banach spaces. 
Then $E$ is ismorphic to finite or countable product
of $l_p(\Lambda_i)$ for suitable sets
$\Lambda_i$ which may be finite. 
\end{claim}

\begin{claim}{Theorem 3.4:}
Let $E$ be the projective limit of the family
$\{l_p(\Lambda_i)|i \in I\}$ which is also complemented 
in product of Banach spaces. Then $E$ is isomorphic to product of 
$\{l_p (\Lambda_i)|i \in J\}$. 
\end{claim}

\begin{claim}{Proof:}
Result follows theorem 2.4 and Lemma 3.3.
\end{claim}

\begin{claim}{Corollary 3.5:}
Let $E$ be complemented subspace of the product of 
$\{l_p(\Gamma_i)|i \in I\}$. Then is also product
of $\{l_p(\Lambda_i)|i \in J\}$. 
\end{claim}

Obviously, if the lemma 3.3, can be proved 
for $c_0(\Gamma_i)$ then the same condusions
can be reached for those spaces, a#!!Ÿûö!<Ò0q3àœ2¹Ð:7�8¹µ·7»¶2²3²�1··:94±:º4··9�7³$0¹°·#À¶4·:44¹�0¹:4±¶2—…….11†…)2³2¹2·1²¹†….2±†…….12³´·=´º2¶´½2¾†….4º2¶­˜—®†…("7¶°·9µ´–!·¶¸62¶²·:2²9º±9¸0±²¹�7³897²:±º7³$4¶12¹:)¸0±²¹–(97±—… ¶²¹&°º4)·±—˜˜œœ˜�œ–˜œ›…….4º2¶­™®†…("7¶°·9µ´�0·2 —'¹:<·9µ´–!·¶¸62¶²·:2²…9º±9¸0±²¹�7³897²:±º7³!0·Each spaces, Trans. Amer. Math. Soc. 316 (1989)
215-231.

\item[3.)]
R. Engelking, General topology (1977) Polish. Scientific Publishers, Warszawa. 

\item[4.)]
J.Gerlits, Continuous function on products of topological spaces. Fund. Math. 
106 (1980), no.1, 67-75.
\end{itemize}



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S. ™nal\\
Department of Mathematics\\
Middle East Technical University\\
06531 Ankara-Turkey\\
e-mail: osul@rorqual.cc.metu.edu.tr
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