Question

In: Advanced Math

Let A be a bounded linear operator on Hilbert space. Show that if R(A) is closed,...

Let A be a bounded linear operator on Hilbert space. Show that if R(A) is closed, so is R(A*)

Solutions

Expert Solution


Related Solutions

Prove that a projection Pe(x) is a linear operator on a Hilbert space, ,H and the...
Prove that a projection Pe(x) is a linear operator on a Hilbert space, ,H and the norm of projection Pe(x) =1 except for trivial case when e = {0}.
7. Show that the dual space H' of a Hilbert space H is a Hilbert space...
7. Show that the dual space H' of a Hilbert space H is a Hilbert space with inner product (', ')1 defined by (f .. fV)1 = (z, v)= (v, z), where f.(x) = (x, z), etc.
Let A be a closed subset of a T4 topological space. Show that A with the...
Let A be a closed subset of a T4 topological space. Show that A with the relative topology is a normal T1
Let T be a linear operator on a finite-dimensional complex vector space V . Prove that...
Let T be a linear operator on a finite-dimensional complex vector space V . Prove that T is diagonalizable if and only if for every λ ∈ C, we have N(T − λIV ) = N((T − λIV )2).
Let V be a Hilbert space. Let f(x) = ∥x∥ for x ∈ V. Using the...
Let V be a Hilbert space. Let f(x) = ∥x∥ for x ∈ V. Using the definition of Frechet differentiation, show that ∇f(x) = x for all x ̸= 0. Furthermore, show that f(x) is not Frechet differentiable at x = 0.
Prove that a subspace of R is compact if and only if it is closed and bounded.
Prove that a subspace of R is compact if and only if it is closed and bounded.
Let A be a subset of all Real Numbers. Prove that A is closed and bounded...
Let A be a subset of all Real Numbers. Prove that A is closed and bounded (I.e. compact) if and only if every sequence of numbers from A has a subsequence that converges to a point in A. Given it is an if and only if I know we need to do a forward and backwards proof. For the backwards proof I was thinking of approaching it via contrapositive, but I am having a hard time writing the proof in...
Let T and S be linear transformations of a vector space V, and TS=ST (a) Show...
Let T and S be linear transformations of a vector space V, and TS=ST (a) Show that T preserves the generalized eigenspace and eigenspace of S. (b) Suppose V is a vector space on R and dimV = 4. S has a minimal polynomial of (t-2)2 (t-3)2?. What is the jordan canonical form of S. (c) Show that the characteristic polynomial of T has at most 2 distinct roots and splits completely.
(10pt) Let V and W be a vector space over R. Show that V × W...
(10pt) Let V and W be a vector space over R. Show that V × W together with (v0,w0)+(v1,w1)=(v0 +v1,w0 +w1) for v0,v1 ∈V, w0,w1 ∈W and λ·(v,w)=(λ·v,λ·w) for λ∈R, v∈V, w∈W is a vector space over R. (5pt)LetV beavectorspaceoverR,λ,μ∈R,andu,v∈V. Provethat (λ+μ)(u+v) = ((λu+λv)+μu)+μv. (In your proof, carefully refer which axioms of a vector space you use for every equality. Use brackets and refer to Axiom 2 if and when you change them.)
Let A and B be two non empty bounded subsets of R: 1) Let A +B...
Let A and B be two non empty bounded subsets of R: 1) Let A +B = { x+y/ x ∈ A and y ∈ B} show that sup(A+B)= sup A + sup B 2) For c ≥ 0, let cA= { cx /x ∈ A} show that sup cA = c sup A hint:( show c supA is a U.B for cA and show if l < csupA then l is not U.B)
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT