Let X be the space of continuous functions : [1, 0) R which have compact support,...

 

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Let X be the space of continuous functions : [1, 0) R which have compact support, that is there exists a compact interval I, of [1, 0) such that x(t) = 0 Vt € [1, 00) \ Iy. Consider X with the norm ||x|| = max |x(t)| and define the mapping T : X→ X as IE[1,00) (Tx)(1) = x(1), for every t e [1, co). (c) Show that T is not bounded, and explain why this is not in contradiction with the Open Mapping Theorem and the Bounded Inverse Theorem. Let X be the space of continuous functions : [1, 0) R which have compact support, that is there exists a compact interval I, of [1, 0) such that x(t) = 0 Vt € [1, 00) \ Iy. Consider X with the norm ||x|| = max |x(t)| and define the mapping T : X→ X as IE[1,00) (Tx)(1) = x(1), for every t e [1, co). (c) Show that T is not bounded, and explain why this is not in contradiction with the Open Mapping Theorem and the Bounded Inverse Theorem.

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