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  3. 1. let x and y be non-linear spaces and t : x --&g

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

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Advance Math

1. Let X and Y be non-linear spaces and T : X -->Y. Provethat if    T is One-to-one then T-1 existon 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, andlet T1 ? B (X, Y) and T2 ? B (Y, Z). letT1T2(x) = T2(T1x)? x ? X.

(i) Prove that T1T2 ? B(X,Y) is also a bounded linear mapping.

(ii) Prove that ??T2T1?? ???T2?? ??T1??

3. If X is an inner product space, then for arbitrary x, y ? X,?< x, y>? ? ??x?? ??y??, provethat the inner product < ? > is a continuousfunction on X by X (Cartesian product) domain.

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