Let X, Y be Banach spaces. Show that T ∈ L(X, Y) is continuous if and...

Let X, Y be Banach spaces. Show that T ∈ L(X, Y) is continuous if and only if T is bounded.

Expert Solution

Colby Messinger answered 5 months ago

1. Let X and Y be non-linear spaces and T : X -->Y. Prove that if ...

1. Let X and Y be non-linear spaces and T : X -->Y. Prove that if T is One-to-one then T-1 exist on R(T) and T-1 : R(T) à X is also a linear map. 2. Let X, Y and Z be linear spaces over the scalar field F, and let T1 ϵ B (X, Y) and T2 ϵ B (Y, Z). let T1T2(x) = T2(T1x) ∀ x ϵ X. (i) Prove that T1T2 ϵ B (X,Y) is also a...

Let (X,dX),(Y,dY ) be metric spaces and f: X → Y be a continuous bijection. Prove...

Let (X,dX),(Y,dY ) be metric spaces and f: X → Y be a continuous bijection. Prove that if (X, dX ) is compact, then f is a homeomorphism. (Hint: it might be convenient to use that a function is continuous if and only if the inverse image of every open set is open, if and only if the inverse image of every closed set is closed).

Problem 16.8 Let X and Y be compact metric spaces and let f: X → Y...

Problem 16.8 Let X and Y be compact metric spaces and let f: X → Y be a continuous onto map with the property that f-1[{y}] is connected for every y∈Y. Show that ifY is connected then so isX.

Let p, q, g : R → R be continuous functions. Let L[y] := y'' +...

Let p, q, g : R → R be continuous functions. Let L[y] := y'' + py' + qy. (i) Explain what it means for a pair of functions y1 and y2 to be a fundamental solution set for the equation L[y] = 0. (ii) State a theorem detailing the general solution of the differential equation L[y] = g(t) in terms of solutions to this, and a related, equation.

(Connected Spaces) (a) Let <X, d> be a metric space and E ⊆ X. Show that...

(Connected Spaces) (a) Let <X, d> be a metric space and E ⊆ X. Show that E is connected iff for all p, q ∈ E, there is a connected A ⊆ E with p, q ∈ E. b) Prove that every line segment between two points in R^k is connected, that is Ep,q = {tp + (1 − t)q | t ∈ [0, 1]} for any p not equal to q in R^k. C). Prove that every convex subset...

please show all the steps and in order Q.5 Let X and Y be continuous rvs...

please show all the steps and in order Q.5 Let X and Y be continuous rvs with the joint pdf f(x, y) = 60(x^2)y, for 0 < x, 0 < y, 0 < x + y < 1 and 0 otherwise. (a) Find E[X + Y ] and E[X − Y ] (b) Find E[XY ] (c) Find E[Y |X = x] and E[X|Y = y]. (d) Find Cov[X, Y ]

please show all steps Q.4 Let X and Y be continuous random variables with the joint...

please show all steps Q.4 Let X and Y be continuous random variables with the joint pdf: f(x, y) = { k(x + y), if (x, y) ∈ 0 ≤ y ≤ x ≤ 1 ; 0 otherwise Answer the question with the equation below please f(x, y) = 2(x + y), for 0 ≤ y ≤ x ≤ 1 and 0 otherwise. (a) Find E[X + Y ] and E[X − Y ] (b) Find E[XY ] (c) Find...

Let X and Y be continuous random variables with E[X] = E[Y] = 4 and var(X)...

Let X and Y be continuous random variables with E[X] = E[Y] = 4 and var(X) = var(Y) = 10. A new random variable is defined as: W = X+2Y+2. a. Find E[W] and var[W] if X and Y are independent. b. Find E[W] and var[W] if E[XY] = 20. c. If we find that E[XY] = E[X]E[Y], what do we know about the relationship between the random variables X and Y?

Let X, Y be two topological spaces. Prove that if both are T1 or T2 then...

Let X, Y be two topological spaces. Prove that if both are T1 or T2 then X × Y is the same in the product topology. Prove or find a counterexample for T0.

1. Let V and W be vector spaces over R. a) Show that if T: V...

1. Let V and W be vector spaces over R. a) Show that if T: V → W and S : V → W are both linear transformations, then the map S + T : V → W given by (S + T)(v) = S(v) + T(v) is also a linear transformation. b) Show that if R: V → W is a linear transformation and λ ∈ R, then the map λR: V → W is given by (λR)(v) =...

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