Real and Complex Analysis Problem 3: Functional Analysis Let H be a separable Hilbert space and T : HH be a compact, self-adjoint operator. Prove that T has an orthonormal basis consisting of eigenvectors of T.

ProofProve in full detail that M2,2, with the standard operations, is a vector space.

True or False? In Exercises 67 and 68, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. a The scalar is an eigenvalue of an nn matrix A when there exists a vector x such that Ax=x. b To find the eigenvalues of an nn matrix A. you can solve the characteristic equation det(IA)=0.

Orthogonal and Orthonormal SetsIn Exercises 1-12, a determine whether the set of vectors in Rnis orthogonal, b if the set is orthogonal, then determine whether it is also orthonormal, and c determine whether the set is a basis for Rn. {(2,4),(2,1)}

Calculus Let B={1,x,sinx,cosx} be a basis for a subspace W of the space of continuous functions and Dx be the differential operator on W. Find the matrix for Dx relative to the basis B. Find the range and kernel of Dx.

Proof Prove that if S1 and S2 are orthogonal subspaces of Rn, then their intersection consists of only the zero vector.

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LINEAR ALGEBRA & FOURIER ANALYSIS