Recent content by SVD

1. Eigenvalues of a compact positive definite operator

eigenvalues of a compact positive definite operator!!! Let A be a compact positive definite operator on Hilbert space H. Let ψ1,...ψn be an orthonormal set in H. How to show that <Aψ1,ψ1>+...+<Aψn,ψn> ≤ λ1(A)+...+λn(A), where λ1≥λ2≥λ3≥..... be the eigenvalues of A in decreasing order. Can...
2. Category and Subcategory

Using the definition given below, I wonder whether we can deduce that for each object A in C', the identity for A in C' coincides with the identity for A in C. Let C' and C be two categories which satisfies that (i)each objects in C' belongs to C (ii)each hom-set in C' is contained in the...
3. Convergence of an Improper Integral

Let f(x) be a continuous functions on [0,∞) and that ∫ |f(t)|^2dt is convergent for 0≤t<∞. Let ∫ |f(t)|^2dt for 0≤t<∞ equals F. Show that lim(σ→∞) ∫(1-x/σ)|f(x)|^2 dx for0≤x≤σ converges to F. I know that it needs to prove that lim(σ→∞) ∫(x/σ)|f(x)|^2 dx for0≤x≤σ converges to 0. Can anyone...
4. How to verify if a mapping is quotient.

Prove or disprove that f is a quotient mapping. f:R^3\{(x1,x2,x3):x1=0}--->R^2 defined by (x1,x2,x3)|->(x2/x1,x3/x1)
5. How do I get the 1st fundamental form on Grassmann Manifold

Consider G(n,m), the set of all n-dimensional subsapce in ℝ^n+m. We define the principal angles between two subspaces recusively by the usual formula. When I see "Differential Geometry of Grassmann Manifolds by Wong", http://www.ncbi.nlm.nih.gov/pmc/articles/PMC335549/pdf/pnas00676-0108.pdf I...