Olet se be u lecally compact topological space. Prove that that GL2) in a cEalyehra (ie. prove (a) Colse) ü a normed *-alyehrn. ( b) Corse) es Complete. (C) lt *Fl = 1F1" ( the cEalyebraic arrom ).
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- Let V be the set of all positive real numbers. Determine whether V is a vector space with the operations shown below. x+y=xyAddition cx=xcScalar multiplication If it is, verify each vector space axiom; if it is not, state all vector space axioms that fail.Let (R,d) be diserete metric space then R is not compact . True or false.??a. Is there any relation between reflexive normed space and a Banach Space? (If yes then prove) b. Give two examples of normed spaces that are not reflexive (with brief reasoning).
- State true or false with a brief justification If the dual X' of a normed linear space X is fininte dimensional, then X is finite dimensionalthe usual metric space defined by d(x,y)= x-y prove the four propertis of metric spaceGive an example with values of a nonstandard operation for M2*2 a vector space that fulfills all axioms
- Prove that topological space E is not homeomorphic to the spaceY = {(x, y) ∈ E^2 : y = ± x} (E represents R equipped with Euclidean distance, E^2 represents R^2 equipped with euclidean distance)Prove that a linear map on a normed vector space is bounded if and only if it iscontinuous.If X is a metric space with induced topology Ƭ, then (X,Ƭ) is Hausdorff. The contrapositive of this theorem must be true:If (X,Ƭ) is not Hausdorff, then X is not a metric space. 1) Consider (ℝ,Ƭ) with the topology induced by the taxicab metric. Using the definition for Hausdorff, give an example of why (ℝ,Ƭ) is Hausdorff. 2) The finite complement topology on ℝ is not Hausdorff. Explain why ℝ with the finite complement topology is non-metrizable.