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 bounded linear mapping.

(ii) Prove that ǀǀT₂T₁ǀǀ ≤ ǀǀT₂ǀǀ ǀǀT₁ǀǀ

3. If X is an inner product space, then for arbitrary x, y ϵ X, ǀ< x, y>ǀ ≤ ǀǀxǀǀ ǀǀyǀǀ, prove that the inner product < ∙ > is a continuous function on X by X (Cartesian product) domain.

Expert Solution

Colby Messinger answered 6 months ago

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