every metric space is normed space? give an example to show tha
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- Determine whether the set R2 with the operations (x1,y1)+(x2,y2)=(x1x2,y1y2) and c(x1,y1)=(cx1,cy1) is a vector space. If it is, verify each vector space axiom; if it is not, state all vector space axioms that fail.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.Find an orthonormal basis for the subspace of Euclidean 3 space below. W={(x1,x2,x3):x1+x2+x3=0}
- Take this test to review the material in Chapters 4 and 5. After you are finished, check your work against the answers in the back of the book. Prove that the set of all singular 33 matrices is not a vector space.Give an example showing that the union of two subspaces of a vector space V is not necessarily a subspace of V.Prove that any reflexive normed space is complete
- Let A be a closed bounded subset of a metric space (X, d). Is A acompact subset? Explain why.Show that the interval (a,b) in R with the discrete metric space is locally compact but not compact.Consider the spaceZ+with the finite complement topology. Is thisspace a Hausdorff space? Is this space aT1space?
