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 . 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 https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429187681/dd6f40d9-b60a-425b-8c69-a9c7f3359253/content/inequ25_1.tif"/> , Ri are Banach spaces, Ri closed subspace in R i+1, i = 1,2,...
There is the projector Pi : R → Ri for each i = 1,2,...
xn → x (xn , x ∈ R, n → ∞) if and only if Pi (xn ) → Pi (x) for each i = 1,2,... (n → ∞)
In particular, Hilbert and Banach spaces are P-spaces.Theorem 1
Let 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:
|Pi (Tx−Ty−(x−y))|≤|Pi (x − y)|. ε(|Pi (x)| + |Pt (y)|), where |ai | is the norm of ai in Ri , 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.
Let i be fixed. Take δ > 0 such that ε(4δ) < 1/2.
Consider the equation
Tx = u, where |Pi (u)| < δ. Construct a sequence xn ∈ R; n = 1,2,... such that x 0 = 0, x n+1=u + xn —Txn . We have
|Pi (X n+1 - Xn )| = |Pi (Xn - X n-1 - Txn + Tx n+1)| ≤
≤ |Pi (xn - x n-1)| · ε(|Pi (xn )| + |Pi (x n-1)|).
So, |Pi (xn )| ≤ |Pi (u)| · (1 + 1/2 + ... + l/2 n-1) and
|Pi (x n+1 - Xn )| ≤ 1/2 n · |Pi (u)|. So, the sequence Pi (xn ), n = 1,2,... converges for any i = 1,2,..., i.e. xn converges in R: xn → x, where Tx = u, i.e. the solution exists.
Besides, we get |Pi (x)| ≤ 2|Pi (u)| < 2δ.
Prove now that the solution is unique. Let an x ∈ R be such that |Pi (x)| < 2δ, Tx = u and xn → x is the above constructed sequence. We have
|Pi (x n+1 - x)| = |Pi (u + xn - Txn - x)| = |Pi (Tx + xn - Txn - x)|≤
≤ |Pi (x - xn )| · ε(|Pi (x)| + |Pi (xn)|). As
|Pi (x)| + |Pi (xn )| ≤ 4δ, then ε(|Pi (x)| + |Pi (xn )|) < 1/2 and
|Pi (x n+1 - x)| ≤ 1/2|Pi (xn - x)|, i.e. |Pi (x n+1 - x)| < 1/2 n |Pi (x)|. Uniqueness is proved. Denote x = Su. So, if |Pi (u)| < δ and |Pi (v)| < δ, then |Pi (Su - Sv)| ≤ 2|Pi (u — v)|. S satisfies the same conditions as T:
|Pi (u - v - (Su - Sv))| < |Pi (Su - Sv)| · ε(|Pi (Su)| + |Pi (Sv)|) ≤ |Pi (u - v)| · ε(|Pi (u)| + |Pi (v)|).
Theorem 1 is proved.
Now we introduce a manifold’s and algebraic structure connected with P-space.Definition 2
Local 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 ∩ Ri contains neighborhood of zero in Ri for each i = 1,2,...
Pi (ab-1 ) = Pi (Pi (a)Pi )-1) for each i = 1,2,..., a, b ∈ G
|Pi (xy)-Pi (x)-Pi (y)| ≤ |Pi (x)|·ε(|Pi (x)| + |Pi (y)|), |Pi (xy)-Pi (x) - Pi (y)| ≤ |Pi (y)| · ε(|Pi (y)|) , if
|Pi (x)| + |Pi (y)| → 0; |·| - norm in Ri 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”.
|Pi (axb - ayb - (x - y))| ≤ |Pi (x - y)| · ε(|Pi (a)| + |Pi (b)| + |Pi (x)| + |Pi (y)|), i=1,2,...;
|Pi ((λx)(λy)) - Pi ((λx) + (λy))| ≤ |Pi (λx)| · ε(|Pi (λx)| + |Pi (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 Pi ((x/n) n ) = Pi ((Pi (x))/n) n ) is defined when |Pi (x)| < δ.
We have now:
|Pi ((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"/> ;
|Pi ((x/n) n )-Pi ((y/n) n )-Pi (x-y))|≤|Pi (x-y)|·ε(|Pi (x)|+|Pi (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) nm = lim m→∞(x/m) mn=lim m→∞((x/m) m ) n , then T(nx) = (Tx) n . Therefore the inverse map S satisfies conditions:
S(xn ) = 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 3
Let 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.
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 Pi (ab -1) = Pi (Pi (a)Pi (b)-1) in P-groups definition are provided by functions, a sufficient number of times differentiable, i e. Pi (ab -1) is a sufficient number of times differentiable functions of Pi (a), Pi (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 |Pi (a)|, |Pi (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.
[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).Corollary 1
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.Theorem 4
If P-group G is Abelian, then its group operation (in “canonical coordinates”) coincides with addition in the corresponding P-space: xy = x+y.
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 3
Algebra Lie T is called P - algebra Lie, if the following conditions are satisfied:
T is P - space
Pi ([ab]) = Pi ([Pi (a), Pi (b)]) for each i = 1,2,... ,
a,b ∊ G,[.,.] - Lie commutator
|Pi ([ab])| ≤ |Pi (a)||Pi (b)| for each i = 1,2,... , a,b ∊ T , |.| - norm in Ri
231In particular, classical finite - dimensional algebras Lie and normed algebras Lie are P - algebras Lie.Theorem 5
Let T be a P-algebra Lie. Consider formal Campbell - Hausdorff series:
Φ(x,y)=Σ((-1) k-1)/k)·1/((p1)!(q1)!...(pk )!(qk ))!.
·1/(p 1+q 1+...+pk +qk ).
[ ... [ [ x , x ] , ... ] , x ] ︸ p 1 , y ] , ... ] , y ] ︸ q 1 , ... , ] x ] , ... ] , x ] ︸ p k , y ] , ... ] , y ] ︸ q 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_3.tif"/> , where the summation is taken over all positive integers and all systems of non-negative integers p 1, q 1,..., pk ,qk with conditions pi +qi , i = 1,..., k. Then the operation x * y = Φ(x, y) defines P-group ΦT with P-space T.
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:
Pi (Φ(x,y)) = Pi (Φ(Pi (x),Pi (y))).
Let |Pi (x)| < α i |Pi (y)| αi . Then
|Pi (Φ(x,y))| ≤ Σ 1/k·1/((p 1)!(q 1)!...(pk )!(qk )! · 1/(p 1+q 1+...pk +qk ).
· (αi ) p 1+q 1+...+pk +qk . So, |Pi (Φ(x,y))| ≤ Σ n≥1 1/nrn (αi ) n ≤ Σ n≥1 rn (αi ) n , where rn = Σ 1/k · 1/(p 1)!(q 1)!...(pk )!(qk )!), where the summation is taken over all systems of non-negative integers p 1, q 1,... ,pk ,qk with conditions p 1+q 1+...+pk + qk = n. The series Σ n≥1 rn (αi ) n is obtained by substitution of the series t = Σ n≥0(1/n!)(αi ) n into theieries s = Σ n≥1(1/n)(t 2 - 1) n .
So, for αi < 0.2 the series Φ(x,y) converges in T.
Theorem 5 is proved.Corollary 2
If 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.