## ABSTRACT

Today modern investigations in various research areas require new mathematical models and algorithms, leading to the infinite-dimensional topological-algebraic structures. For example, for control systems in turbomachinery it is necessary to describe the position of the compressor design (or operating) point as a function in some mfinite-dimensional space. In Genetics it is also necessary to consider DNA sequences, Gene expression data as elements of some infinite-dimensinal manifold in some infinite-dimensional space. This paper describes in details infinite-dimensional technique based on the concept of P-spaces, which was introduced in [1]. Definition 1

Linear topological space R over field of real (complex) numbers is called P - space, if the following conditions are satisfied:

R
⊃
∪
i
=
1
∞
R
i
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, R_{i}
are Banach spaces, R_{i}
closed subspace in R
_{
i+1}, i = 1,2,...

There is the projector P_{i}
: R → R_{i}
for each i = 1,2,...

x_{n}
→ x (x_{n}
, x ∈ R, n → ∞) if and only if P_{i}
(x_{n}
) → P_{i}
(x) for each i = 1,2,... (n → ∞)

In particular, Hilbert and Banach spaces are P-spaces.

Theorem 1Let R be a P-space, and let for a certain neighbourhood of 0 ∈ R a map T of the space onto itself be defined, satisfying the conditions:

|P_{i}
(Tx−Ty−(x−y))|≤|P_{i}
(x − y)|. ε(|P_{i}
(x)| + |P_{t}
(y)|), where |a_{i}
| is the norm of a_{i}
in R_{i}
, i = 1,2,...ε(ξ) is a real positive numerical function such that ε(ξ)→ 0 when ξ → 0. Note that everywhere further we on ε(ξ) will denote any such function. Then

228 T is a local homeomorphism. Moreover, its inverse map also satisfies same contitions.

Proof:

Let i be fixed. Take δ > 0 such that ε(4δ) < 1/2.

Consider the equation

Tx = u, where |P_{i}
(u)| < δ. Construct a sequence x_{n}
∈ R; n = 1,2,... such that x
_{0} = 0, x
_{
n+1}=u + x_{n}
—Tx_{n}
. We have

|P_{i}
(X
_{
n+1} - X_{n}
)| = |P_{i}
(X_{n}
- X
_{
n-1} - Tx_{n}
+ Tx
_{
n+1})| ≤

≤ |P_{i}
(x_{n}
- x
_{
n-1})| · ε(|P_{i}
(x_{n}
)| + |P_{i}
(x
_{
n-1})|).

So, |P_{i}
(x_{n}
)| ≤ |P_{i}
(u)| · (1 + 1/2 + ... + l/2^{
n-1}) and

|P_{i}
(x
_{
n+1} - X_{n}
)| ≤ 1/2^{
n
} · |P_{i}
(u)|. So, the sequence P_{i}
(x_{n}
), n = 1,2,... converges for any i = 1,2,..., i.e. x_{n}
converges in R: x_{n}
→ x, where Tx = u, i.e. the solution exists.

Besides, we get |P_{i}
(x)| ≤ 2|P_{i}
(u)| < 2δ.

Prove now that the solution is unique. Let an x ∈ R be such that |P_{i}
(x)| < 2δ, Tx = u and x_{n}
→ x is the above constructed sequence. We have

|P_{i}
(x
_{
n+1} - x)| = |P_{i}
(u + x_{n}
- Tx_{n}
- x)| = |P_{i}
(Tx + x_{n}
- Tx_{n}
- x)|≤

≤ |P_{i}
(x - x_{n}
)| · ε(|P_{i}
(x)| + |P_{i}
(x_{n})|). As

|P_{i}
(x)| + |P_{i}
(x_{n}
)| ≤ 4δ, then ε(|P_{i}
(x)| + |P_{i}
(x_{n}
)|) < 1/2 and

|P_{i}
(x
_{
n+1} - x)| ≤ 1/2|P_{i}
(x_{n}
- x)|, i.e. |P_{i}
(x
_{
n+1} - x)| < 1/2^{
n
}|P_{i}
(x)|. Uniqueness is proved. Denote x = Su. So, if |P_{i}
(u)| < δ and |P_{i}
(v)| < δ, then |P_{i}
(Su - Sv)| ≤ 2|P_{i}
(u — v)|. S satisfies the same conditions as T:

|P_{i}
(u - v - (Su - Sv))| < |P_{i}
(Su - Sv)| · ε(|P_{i}
(Su)| + |P_{i}
(Sv)|) ≤ |P_{i}
(u - v)| · ε(|P_{i}
(u)| + |P_{i}
(v)|).

Theorem 1 is proved.

Now we introduce a manifold’s and algebraic structure connected with P-space.

Definition 2Local topological group G is called P - group, if the following conditions are satisfied:

G ⊂ R with 0 = e , R is P - space , 0 is zero in R, e is unity element of G.

G ∩ R_{i}
contains neighborhood of zero in R_{i}
for each i = 1,2,...

P_{i}
(ab^{-1}
) = P_{i}
(P_{i}
(a)P_{i}
)^{-1}) for each i = 1,2,..., a, b ∈ G

|P_{i}
(xy)-P_{i}
(x)-P_{i}
(y)| ≤ |P_{i}
(x)|·ε(|P_{i}
(x)| + |P_{i}
(y)|), |P_{i}
(xy)-P_{i}
(x) - P_{i}
(y)| ≤ |P_{i}
(y)| · ε(|P_{i}
(y)|) , if

|P_{i}
(x)| + |P_{i}
(y)| → 0; |·| - norm in R_{i}
for each i = 1,2,..., x,y ∈ G

In particular, classical finite - dimensional groups Lie and Birkhoff = groups are P - groups.

<target id="page_229" target-type="page">229</target> <xref ref-type="boxed-text" rid="the25_2">Theorem 2</xref> Local topological P-group <italic>G</italic> has “canonical coordinates”.Proof:

We have:

|P_{i}
(axb - ayb - (x - y))| ≤ |P_{i}
(x - y)| · ε(|P_{i}
(a)| + |P_{i}
(b)| + |P_{i}
(x)| + |P_{i}
(y)|), i=1,2,...;

|P_{i}
((λx)(λy)) - P_{i}
((λx) + (λy))| ≤ |P_{i}
(λx)| · ε(|P_{i}
(λx)| + |P_{i}
(λy)|), i = 1,2,...; lim_{
λ→0}((λx)(λy))/λ = x + y.

For any i = 1,2,... there exists such δ > 0 that the expression P_{i}
((x/n)
^{n}
) = P_{i}
((P_{i}
(x))/n)
^{n}
) is defined when |P_{i}
(x)| < δ.

We have now:

|P_{i}
((x/n)
^{n}
-(y/n)
^{n}
-(x-y))|≤

≤ Σ k = 1 n | P i ( ( x / n ) k − 1 ( x / n ) ( y / n ) n − k − ( x / n ) k − 1 ( y / n ) ( y / n ) n − k − ( x / n − y / n ) ) | ≤ ≤ | P i ( x / n − y / n ) | ⋅ Σ k = 1 n ϵ ( | P i ( x / n ) | + | P i ( y / n ) | + | P i ( ( x / n ) k − 1 ) | + | P i ( ( y / n ) n − k ) | ) https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429187681/dd6f40d9-b60a-425b-8c69-a9c7f3359253/content/inequ25_2.tif"/> ;

|P_{i}
((x/n)
^{n}
)-P_{i}
((y/n)
^{n}
)-P_{i}
(x-y))|≤|P_{i}
(x-y)|·ε(|P_{i}
(x)|+|P_{i}
(y)|) for any n and any i = 1,2,.... So, lim_{
n→∞}(x/n)
^{n}
exists and let lim_{
n→∞}(x/n)
^{n}
= Tx. Map T meets conditions of Theorem 1. Besides, as lim_{
m→∞}(nx/m)
^{m}
= lim_{
m→∞}(nx/nm)
^{n}m = lim_{
m→∞}(x/m)
^{m}n=lim_{
m→∞}((x/m)
^{m}
)
^{n}
, then T(nx) = (Tx)
^{n}
. Therefore the inverse map S satisfies conditions:

S(x^{n}
) = nS(x),n = 0,±1,±2,.... Let x,y,z ∊ R(G) and let Sx = x′,Sy = y′, Sz = z′. Define x′ * y′ = z′ if and only if xy = z, and define (x′)^{*-1} = y′ if and only if x
^{-1} = y. Thus we get a new local topological group G′ with group operation *, locally isomorphic to G in a neighborhood of zero in P-space R with conditions (x′)^{
n
}
nx′.

Theorem 2 is proved.

Theorem 3Let φ : G
_{1} → G
_{2} be a continuous homomorphism of P-groups G
_{1} and G
_{2} - both in “canonical coordinates”. Then φ coincides in some neighborhood of zero with the linear continuous operator Φ : R
_{1} → R
_{2} for corresponding P-spaces.

Proof:

We have for integers m,n,n ≠ 0:

φ((m/n)x) = mφ(x/n),φ((n/n)x) = nφ(x/n),φ((m/n)x) = m/nφ(x). So, φ((λ)x) = λφ(x) for any λ, if the expressions in both - left and right - sides are defined. φ(x+y) = lim_{
λ→0}(((λx)(λy))/λ) = lim_{
λ→0}(φ(λx)φ(λy))/λlim
_{
λ→0}(λφ(x)λφ(x)+φ(y). It is obvious that φ coincides with some Φ.

Theorem 3 is proved.

We assume now that expressions P_{i}
(ab
^{-1}) = P_{i}
(P_{i}
(a)P_{i}
(b)^{-1}) in P-groups definition are provided by functions, a sufficient number of times differentiable, i e. P_{i}
(ab
^{-1}) is a sufficient number of times differentiable functions of P_{i}
(a), P_{i}
(6). It is possible to prove this statement (not to assume), but this proof is above the scope of this paper. So, we have: ab = a+b+A(a, b) +..., where a, b ∈ G, A is a bilinear operator, and dots denote terms of higher then second order of smallness as related 230to |P_{i}
(a)|, |P_{i}
(b)| for any i = 1,2,....

Let [a,b] = A(a, b) — A(b,a). It can be easily seen that the operation [,] may be extended to the whole of P-space R and is Lie commutator operation in R. So, R becomes Lie algebra LG.

We have:

[x,y]=lim_{
λ→0}((λx)(λy)-(λy)(λx))/λ
^{2});

[x,y]=lim_{
λ→0}((λx)(λy)(λx)^{-1}-(λy))/λ
^{2});

[x,y]=lim_{
λ→0}((λx)(λy)(λx)^{-1}-(λy)^{-1})/λ
^{2}).

If two P-groups G
_{1} and G
_{2} are locally isomorphic, then corresponding Lie algebras LG
_{1} and LG
_{2} are isomorphic.

Let H is closed subgroup of G, containing with any element x its one-parameter subgroup (a line) g(t): x ∊ g(t). Then Lie algebra LH is subalgebra of LG.

Let H is closed normal subgroup of G, containing with any element x its one-parameter subgroup (a line) g(t): x ∊ g(t). Then Lie algebra LH is a closed ideal of LG.

Let φ : G
_{1} → G
_{2} be a continuous homomorphism of P-groups G
_{1} and G
_{2}. Then φ coincides in some neighborhood of zero with the continuous homomorphism Φ : LG
_{1} → LG
_{2} for corresponding Lie algebras.

If P-group G is Abelian, then its group operation (in “canonical coordinates”) coincides with addition in the corresponding P-space: xy = x+y.

Proof:

As xyx
^{-1} = y, then x(λy)x
^{-1} = λy. In the same way yxy
^{-1} = x,

y(μx)y
^{-1} = μx, i.e.

(λy)(μx) = (μx)(λy). We have:

((x/n)(y/n))
^{n}
= (x/n)
^{n}
(y/n)
^{n}
, i.e.

xy = ((x/n)(y/n))/(1/n) = x + y.

Theorem 4 is proved.

Now we present an internal definition for Lie algebra LG which we have developed.

Definition 3Algebra Lie T is called P - algebra Lie, if the following conditions are satisfied:

T is P - space

P_{i}
([ab]) = P_{i}
([P_{i}
(a), P_{i}
(b)]) for each i = 1,2,... ,

a,b ∊ G,[.,.] - Lie commutator

|P_{i}
([ab])| ≤ |P_{i}
(a)||P_{i}
(b)| for each i = 1,2,... , a,b ∊ T , |.| - norm in R_{i}

231In particular, classical finite - dimensional algebras Lie and normed algebras Lie are P - algebras Lie.

Theorem 5Let T be a P-algebra Lie. Consider formal Campbell - Hausdorff series:

Φ(x,y)=Σ((-1)^{
k-1})/k)·1/((p1)!(q1)!...(p_{k}
)!(q_{k}
))!.

·1/(p
_{1}+q
_{1}+...+p_{k}
+q_{k}
).

[
...
[
[
x
,
x
]
,
...
]
,
x
]
︸
p
1
,
y
]
,
...
]
,
y
]
︸
q
1
,
...
,
]
x
]
,
...
]
,
x
]
︸
p
k
,
y
]
,
...
]
,
y
]
︸
q
k
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, where the summation is taken over all positive integers and all systems of non-negative integers p
_{1}, q
_{1},..., p_{k}
,q_{k}
with conditions p_{i}
+q_{i}
, i = 1,..., k. Then the operation x * y = Φ(x, y) defines P-group ΦT with P-space T.

Proof:

First of all we have:

Φ(x, 0) = 0, Φ(x, -x) = 0, Φ(x, Φ(y, z)) = Φ(Φ(x, y), z). Take now 0 ∈ T as e ∈ ΦT and let for any x ∈ T be x
^{—1} = —x. For any i = 1,2,... we have:

P_{i}
(Φ(x,y)) = P_{i}
(Φ(P_{i}
(x),P_{i}
(y))).

Let |P_{i}
(x)| < α_{
i
}|P_{i}
(y)| α_{i}
. Then

|P_{i}
(Φ(x,y))| ≤ Σ 1/k·1/((p
_{1})!(q
_{1})!...(p_{k}
)!(q_{k}
)! · 1/(p
_{1}+q
_{1}+...p_{k}
+q_{k}
).

· (α_{i}
)^{
p
1+q
1+...+pk
+qk
}. So, |P_{i}
(Φ(x,y))| ≤ Σ_{
n≥1} 1/nr_{n}
(α_{i}
)
^{n}
≤ Σ_{
n≥1}
r_{n}
(α_{i}
)^{
n
}, where r_{n}
= Σ 1/k · 1/(p
_{1})!(q
_{1})!...(p_{k}
)!(q_{k}
)!), where the summation is taken over all systems of non-negative integers p
_{1}, q
_{1},... ,p_{k}
,q_{k}
with conditions p
_{1}+q
_{1}+...+p_{k}
+ q_{k}
= n. The series Σ_{
n≥1}
r_{n}
(α_{i}
)
^{n}
is obtained by substitution of the series t = Σ_{
n≥0}(1/n!)(α_{i}
)
^{n}
into theieries s = Σ_{
n≥1}(1/n)(t
^{2} - 1)
^{n}
.

So, for α_{i}
< 0.2 the series Φ(x,y) converges in T.

Theorem 5 is proved.

Corollary 2If two P-algebras T
_{1} and T
_{2} are isomorphic, then corresponding P-groups ΦT
_{1} and ΦT
_{2} are locally isomorphic.

Let U is closed subalgebra of T. Then P-group ΦU is subgroup of ΦT.

Let U is closed ideal of T. Then P-group ΦU is normal subgroup of ΦT.

Let ψ : T
_{1} → T
_{2} be a continuous homomorphism of P-algebras T
_{1} and T
_{2}. Then ψ coincides in some neighborhood of zero with the continuous homomorphism Ψ : ΦT
_{1} - ΦT
_{2} for corresponding P-groups.

I’d like to thank Ilya Markevich and Professor Pavel Krutitskii for their help. And also I would like to thank Professor Joji Kajiwara for his proposal to write this paper.