Let be a topological space, let and be paths in such that . Show that defined by is a path in
Let be a topological space, let and be paths in such that . Show that defined...
Let V be a Hilbert space. Let S1 and S2 be two hyperplanes in V defined by Let be given. We consider the projection of y onto , i.e., the solution of (1) (a) Prove that is a plane, i.e., if , then for any . (b) Prove that z is a solution of (1) if and only if and (2) (c) Find an explicit solution of (1). ( d) Prove the solution found in part (c) is unique. We...
Let be an orthonormal set of a Hilbert space. Let and be two vectors in H. Show that converges absolutely, and that We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
Let be a metric space and let be the topology on induced by , and let be a compact space. Prove that is compact. (x, d) We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageAj,i=1,2,... na1 An
Let be an inner product space (over or ), and . Prove that is an eigenvalue of if and only if (the conjugate of ) is an eigenvalue of (the adjoint of ). We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageTEL(V) We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to...
Let T be defined by a) Show that the and that the eigenvalues are given by 1 and -1 b) Determine the eigenvectors We were unable to transcribe this imageTEB TEB
Let , and let be a polynomial. Show that if is an eigenvalue of , then is an eigenvalue of . Hint: this follows from the more precise statement that if is a non-zero eigenvector for for the eigenvalue , then is also an eigenvector for for the eigenvalue . Prove this. TEL(V) PEPF) We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were...
Let X be a banach space such that X= C([a,b]) where - ab+ with the sup norm. Let x and f X. Show that the non linear integral equation u(x) = (sin u(y) dy + f(x) ) has a solution u X. (the integral is from a to b). We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe...
Let a,b and c be real numbers and consider the function defined by . For which values of a,b, and c is f one-to-one and or onto ? Show all work. f:R→R We were unable to transcribe this imageWe were unable to transcribe this image f:R→R
Show that the tensor defined from the metric tensor satisfies the symmetry property Evaluate the contracted tensors and in four dimensions and in general n dimensions We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
Let be independent, identically distributed random variables with . Let and for , . (a) Show that is a martingale. (b) Explain why satisfies the conditions of the martingale convergence theorem (c) Let . Explain why (Hint: there are at least two ways to show this. One is to consider and use the law of large numbers. Another is to note that with probability one does not converge) (d) Use the optional sampling theorem to determine the probability that ever attains...