Consider the Hilbert space ℓ2 of square-summable complex sequences(a1, a2, . . . ).You may assume that the subset S = {e1, e2, . . . } with e1 = (1, 0, 0, . . . ), e2 = (0, 1, 0, . . . ),. . . , is a Schauder basis for ℓ2.Let T ∶ ℓ2 Ð→ ℓ2 be a linear operator

Consider the Hilbert space ℓ2 of square-summable complex sequences(a1, a2, . . . ).You may assume that the subset S = {e1, e2, . . . } with e1 = (1, 0, 0, . . . ), e2 = (0, 1, 0, . . . ),. . . , is a Schauder basis for ℓ2.Let T ∶ ℓ2 Ð→ ℓ2 be a linear operator.(a) Show that T is a bounded linear operator if and only if the sequence (∥T (en)∥) isbounded.(b) IfT (ej ) =∞∑n=1cjn en,give a condition on the coefficients cjn that is necessary and sufficient for T to beself-adjoint.

Let S be a subset of a Hilbert space H. Prove that Span(S) is dense in H if and only ifS⊥ = 0.

Let T be an orthogonal operator on ℝ2 and let A be its matrix representation with respect to the standard ordered basis of ℝ2. Which one of the following st

T: P₂→ P₂ is a linearoperator. Determine whether the set\{T(1), T(x), T(x ^ 2)\} is a basis for P2. 19. . T(a * x ^ 2 + bx + c) = (a + b + c) * x ^ 2 + (a + b) * x + a

Which operator performs pattern matching?BETWEEN operatorLIKE operatorEXISTS operatorHAVING operator