Topological vector space over the quaternions

Abstract

Lex X be a vector space over lH. A map ǁ.ǁ : X →ℝ is said to be a paranorm on x if and only if (1) ǁ0 ǁ = 0 (2) For all x ∈ X, ǁ-xǁ= ǁxǁ (3) For all x, y ∈ X, ǁx+yǁ≤ ǁxǁ+ǁyǁ (4) If (tn)n∈ ℕ is a sequence of elements in lH such that t n →t and (xn)n∈ ℕ is a sequence of elements in x such that ǁXn-Xǁ →0 then ǁtn-Xn-txǁ →0 We call (x, ǁ.ǁ) a paranorm space over lH Let x be a vector space over lH. A map ǁ.ǁ : x → ℝ is said to be a seminorm on x if and only if (1) For all x∈ x and t , ∈ ℍ, ǁ tx ǁ= ⃒t⃒‖x‖ (2) For all x, y∈ x , x+y ≤ ‖x‖ + ‖y‖ We call (x, ǁ.ǁ ) a seminorm space over lH. A topological vector space X over lH (T V S (lH)) is a topological space and a vector space over lH such that the vector operations are continuous. Theorem Let (X, p) be a seminormed space over lH. Let f be a linear functional defined only on a vector subspace S of x and such that ⃒f (x)⃒≤ p(x) for all x ∈ s. Then f can be extended to F ∈ x # with ⃒f (x)⃒≤ p(x) for all x∈ x. Theorem Let (x, ǁ.ǁ ₁ ) and (y, ǁ.ǁ ₂ ) be seminormed space over lH and let T: x →Y be a linear map. Then the following are equivalent : (1) T is continuous at some a ∈x (2) T is continuous on X. (3) T is bounded on the unitdisc, i.e. ‖T ‖<∞. (4) There exists an M ∈ ℝ such that ‖T(x)‖₂≤M‖x‖ ₁ for all X ∈X. Theorem Let (X, T) be a first countable T V S (lH). Then there exists a paranorm ǁ.ǁ on X such T = T where T is the topology induced by ǁ.ǁ Theorem Every T V S (lH) is a completely regular topological space. Theorem Let X be a T V S (lH), f∈ x# and assume that ker f is closed. Then f is continuous on X. Theorem (x, ‖.‖ ) be a seminormed space over IH. Then the quotient space of X is also a seminormed space over IH. Theorem Let X be an n-dimensional separated T V S (IH), n < ∞. Then X is linearly homeomorphic with IHⁿ. Theorem Let X be a separated T V S (IH) which has a totally bounded neighborhood U of 0. Then X is finite dimensional. Theorem Let X be a T V S (IH). Then K C X is compact if and only if K is complete and totally bounded. Theorem (Open mapping theorem) Let X, Y be F S (IH)’s and let f : X→Y be linear, continuous and onto. Then f is open. Theorem (Closed graph theorem) Let X, Y be FS (IH)’s. Let f : X→Y be a linear map with a closed graph G. Then f is continuous. Theorem Every basis of a F S (IH) is a Schauder basis. Theorem Let (X, P) be a locally convex space over IH> and f∈s where S is a vector subspace of X. Then there exists an F∈X such that F = f on s. Theorem Let ɸ be a collection of locally convex topologies on a vactor space X over IH. Then f ∈(x, v ɸ ) if and only if there exists T₁, T₂, … T ∈ɸ ; g₁, g₂, …., gn∈X # such that each gi∈ (x,Tᵢ) and f =Σⁿgᵢ₌ ᵢ₌₁