P-Spaces

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 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"/> , 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 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:

|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 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 ∩ 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) 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.

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).

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.

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 3

Algebra 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 5

Let 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 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,..., 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+...+p_k +q_k} . 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 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.