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MATH 7280 Operator Theory-Spring 2006 Instructor: Marian Bocea Homework Assignment # 1 (Due February 17, 2006) 1. Let E be a normed space, and f : E → R be a linear map. Show that the hyperplane [f = α] := {x ∈ E : f (x) = α} is closed if and only if f is continuous. 2. Let c0 := {{xn }n∈N : xn → 0 as n → ∞} be the vector space of sequences of real numbers converging to 0 endowed with the norm k{xn }n∈N k := sup |xn |. N n∈ Consider f : c0 → R given by f ({xn }n∈N ) := Show that ∞ P n=1 xn . n2 (i) f is linear and continuous (ii) the supremum in kf kc00 = sup {hf, xi : x ∈ c0 , kxk ≤ 1} is not attained. 3. Give an example of two linear continuous maps f1 , f2 : c0 → R, f1 6= f2 , and an element x0 ∈ c0 such that kf1 kc00 = kf2 kc00 = kx0 k and hf1 , x0 i = hf2 , x0 i = kx0 k2 . 1