1. Introduction
For classical results on symmetric polynomials on finite dimensional spaces, we refer to [
1,
2,
3]. Symmetric polynomials on infinite dimensional Banach spaces were studied, firstly, by Nemirovski and Semenov in [
4]. In particular, in [
4] the authors constructed a countable algebraic basis (see definition below) of the algebra of symmetric continuous real-valued polynomials on the real Banach space
and a finite algebraic basis of the algebra of symmetric continuous real-valued polynomials on the real Banach space
where
In [
5], these results were generalized to separable sequence real Banach spaces with symmetric basis (see, e.g., ([
6], Definition 3.a.1, p. 113) for the definition of a Banach space with symmetric basis) and to separable rearrangement invariant function real Banach spaces (see, e.g., ([
7], Definition 2.a.1, p. 117) for the definition of a rearrangement invariant function Banach space) resp. In [
8], it was shown that there are only trivial symmetric continuous polynomials on the space
Consequently, the results of [
5] cannot be generalized to nonseparable sequence Banach spaces. The most general approach to the studying of symmetric functions on Banach spaces was introduced in [
9,
10,
11,
12,
13].
Note that the existence of a finite or countable algebraic basis in some algebra of symmetric continuous polynomials gives us the opportunity to obtain some information or, even, to describe spectra of topological algebras of symmetric holomorphic functions, which contain the algebra of symmetric continuous polynomials as a dense subalgebra. For example, in [
14], the authors constructed an algebraic basis of the algebra of symmetric continuous complex-valued polynomials on the complex Banach space
of complex-valued Lebesgue measurable essentially bounded functions on
This result gave us the opportunity to describe the spectrum of the Fréchet algebra
of symmetric analytic entire functions, which are bounded on bounded sets on the complex Banach space
(see [
14]) and to show that the algebra
is isomorphic to the algebra of all analytic functions on the strong dual of the topological vector space of entire functions on the complex plane
(see [
15]).
In [
16,
17], there were constructed algebraic bases of algebras of symmetric continuous polynomials on Cartesian powers of complex Banach spaces
and
of all complex-valued Lebesgue integrable in a power
p functions on
and
resp., where
These results gave us the opportunity to represent Fréchet algebras of symmetric entire analytic functions of bounded type on these Cartesian powers as Fréchet algebras of entire analytic functions on their spectra (see [
18]).
The spectra of algebras with countable algebraic bases and completions of such algebras also were studied in [
19,
20,
21]. Symmetric analytic functions of unbounded type were studied in [
22,
23,
24,
25]. Applications of symmetric analytic functions to the spectra of linear operators were introduced in [
26].
Symmetric polynomials and symmetric holomorphic functions on spaces
were studied by a number of authors [
22,
27,
28,
29,
30,
31,
32,
33,
34,
35,
36,
37,
38,
39,
40,
41] (see also the survey [
42]). Symmetric polynomials and symmetric holomorphic functions on Cartesian powers of spaces
were studied in [
43,
44,
45,
46,
47]. In particular, in [
46] there was constructed a countable algebraic basis of the algebra of all symmetric continuous complex-valued polynomials on the Cartesian power of the complex Banach space
This result was generalized to the real case in [
47]. In this work, we generalize the results of the work [
46] to the algebra of symmetric continuous polynomials on the arbitrary Cartesian product
2. Preliminaries
We denote by and the set of all positive integers and the set of all nonnegative integers resp.
2.1. Polynomials
Let
X be a complex Banach space with norm
A function
is called an
N-homogeneous polynomial if there exist
and an
N-linear form
such that
P is the restriction of
to the diagonal, i.e.,
for all
.
A function
which can be represented in the form
where
is a constant function on
X and
is a
j-homogeneous polynomial for every
which is called a polynomial of degree at most
It is known that a polynomial
is continuous if and only if its norm
is finite. Consequently, if
is a continuous
N-homogeneous polynomial, then we have
for every
For details on polynomials on Banach spaces, we refer the reader to [
48] or [
49,
50].
2.2. Algebraic Combinations and Algebraic Basis
Let functions
act from
T to
where
T is an arbitrary nonempty set. If there exists a polynomial
such that
for every
then the function
f is called an algebraic combination of functions
A set
is called algebraically independent if the fact that
for all
implies that the polynomial
Q is identically equal to zero. An infinite set of functions is called algebraically independent if every finite subset is algebraically independent. Note that the algebraic independence implies the uniqueness of the representation in the form of an algebraic combination.
Let be an algebra of functions. A subset of is called an algebraic basis of if each element of can be uniquely represented as an algebraic combination of some elements of
2.3. Symmetric Polynomials on the Space
For let be the space of all sequences where and Note that is isomorphic to Let
A function
is called symmetric if
for every
and for every bijection
where
For
let us define a polynomial
by
where
Let M be a nonempty finite subset of . Let be the vector space of all functions . Elements of the space can be considered as -dimensional complex vectors , indexed by elements of where is the cardinality of Thus, is isomorphic to The space we endow with norm where
For a nonempty finite subset
M of
let us define a mapping
by
where
Theorem 1 ([
46], Theorem 9).
Let be a symmetric N-homogeneous polynomial. Let There exists a polynomial such that where the mapping is defined by (4). We shall use the following lemma.
Lemma 1 ([
46], Lemma 11).
Let and be an orthogonal projection: Let and for every open set a set is unbounded. If polynomial is bounded on then Q does not depend on 3. The Main Result
Let
and
We shall consider the Cartesian power
of the complex spaces
as the space of all sequences
where
for
, such that the sequence
belongs to
for every
. We endow
with norm
where
is the norm of the space
. Note that
is a dense subspace of
Analogically to the definition of symmetric functions on
a function
is called symmetric if
for every
and for every bijection
where
Let
be such that
. Let us define a polynomial
by
Note that the polynomial is symmetric. Let us show that is well-defined and continuous.
Lemma 2. Let be such that Let be such that Then, Proof. Since
taking into account the inequality
This completes the proof. □
Proposition 1. The polynomial , defined by (6), is well-defined and Proof. Let us show that
is well-defined. Let
be of the form (
5). Since the series
are convergent, it follows that there exists
such that
for every
Therefore, for
taking into account the inequality
by Lemma 2,
Therefore, the series is absolutely convergent. Thus, is well-defined.
Let us show that
Let
be such that
Then,
for every
Therefore, for every
by Lemma 2,
Thus, This completes the proof. □
For arbitrary
, we set
For
let
Note that for every
For every
and
, we set
We set
For
let
where
for
Let us define a partial order on in the following way. For , we set if and only if there exists such that for every We write if and
By ([
46], Proposition 3), for every
where by the definition, the product of an empty set of multipliers is equal to
In particular,
For
and
let
It can be easily verified that
where
Note that
For let and
Lemma 3. Let be such that If for some then Consequently,
Proof. Since there exists such that for every Consequently, if then
If then Therefore, cannot be equal to zero. Consequently, Thus, This completes the proof. □
Lemma 4. Let be such that and Then, Proof. By Lemma 3, On the other hand, i.e., Therefore, Consequently, if and only if
Since
there exists
such that
for every
Since
if and only if
it follows that
for every
such that
Since
there exists
such that
Consequently,
This completes the proof. □
For
and
let
Note that ([
46], Theorem 6) implies the following theorem.
Theorem 2. Let M be a finite non-empty subset ofThen,
- (i)
there existssuch that, for everythere existssuch thatand
- (ii)
there exists a constantsuch that ifthen
By Theorem 2, for
, there exists
such that
contains the open unit ball of the space
with the norm
where
Since for every it follows that Consequently, also contains the open unit ball of the space
Proposition 2. For , let be a polynomial on If q is bounded on then q does not depend on where is such that and
Proof. Let
be such that
and
Let
and
be an orthogonal projection, defined by
Let us show that, for every ball
with center
and radius
such that
a set
is unbounded. Since
there exists
such that
For
we set
where
is defined by (
10) and
for
and
where
Since
it follows that
Consequently, the value
is finite, where
is the Riemann zeta function. Choose
such that
Let
Let us show that
By (
7),
By the triangle inequality,
Since it follows that Hence,
Note that, for arbitrary
by (
8),
Consequently, taking into account (
8), we have
Let us show that
For
such that
by (
11),
therefore, by (
17),
Let
be such that
Consider the case
and
Since
it follows that
Consider the case
and
Since
it follows that
By Lemma 4, since
and
Thus, for
such that
we have
By (
17), taking into account the equality
we have
Since
it follows that
By (
2), taking into account the inequality
we have
Since
for every
we have
Therefore,
Since
it follows that
therefore,
By (
12),
therefore, by (
17),
as
Hence,
is unbounded. By Lemma 1,
q does not depend on
This completes the proof. □
Theorem 3. Let be an N-homogeneous symmetric continuous polynomial. If then Otherwise, there exists the polynomial such that whereand is defined by Proof. Let
be the restriction of
P to
Note that
is a symmetric
N-homogeneous polynomial. By Theorem 1, there exists a unique polynomial
such that
Since
P is continuous,
P is bounded on
defined by (
16). Consequently,
is bounded on
Therefore,
q is bounded on
Note that
where
is defined by (
15).
Let us prove that q does not depend on arguments such that by induction on By Proposition 2, for , we have that does not depend on arguments such that and Suppose the statement holds for where i.e., does not depend on arguments such that and Then, the restriction of q to by Proposition 2, does not depend on such that and Hence, q does not depend on such that
Consider the case
In this case,
for every
Consequently,
q is constant. Therefore, taking into account (
19),
is constant. Since
is an
N-homogeneous polynomial, where
it follows that
is identically equal to zero. By the continuity of
taking into account that
is the restriction of
P to the dense subspace
of the space
the polynomial
P is identically equal to zero.
Consider the case
In this case,
Since
q does not depend on
such that
the equality (
19) implies the following equality:
where
is the restriction of
q to
which is the subspace of
Let us show that
Let
Since
is dense in
there exists the sequence
which is convergent to
Since
is continuous and
is the restriction of
it follows that
for every
Therefore,
Since
is the polynomial on a finite dimensional space, it follows that
is continuous. Consequently,
On the other hand, since
P is continuous, taking into account (
20), we have
Therefore, Thus, This completes the proof. □
Proposition 3. The set of polynomialsis algebraically independent. Proof. By ([
46], Theorem 10), the set of polynomials
is an algebraic basis of the algebra of all symmetric continuous complex-valued polynomials on
Consequently, this set of polynomials is algebraically independent. Since every subset of an algebraically independent set is algebraically independent, the set of polynomials
is algebraically independent. Since
is the restriction of
for every
such that
it follows that the set (
21) is algebraically independent. This completes the proof. □
Theorem 4. The set of polynomials (21) is an algebraic basis of the algebra of all symmetric continuous complex-valued polynomials on Proof. Let
be a symmetric continuous complex-valued polynomial of degree at most
where
. Then,
where
and
is a
j-homogeneous polynomial for every
By the Cauchy Integral Formula for holomorphic functions on Banach spaces (see, e.g., ([
48], Corollary 7.3, p. 47)),
for every
and
where
Consequently,
is symmetric and continuous for every
Therefore, by Theorem 3,
can be represented as an algebraic combination of elements of the set (
21) for every
Consequently,
P can be represented as an algebraic combination of elements of the set (
21). Since, by Proposition 3, the set (
21) is algebraically independent, the above-mentioned representation of
P as an algebraic combination of elements of (
21) is unique. Thus, every symmetric continuous complex-valued polynomial on
can be uniquely represented as an algebraic combination of elements of the set (
21). This completes the proof. □