function by where Let (X, d) b a compact metric space and a > 0. We...
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function by where Let (X, d) b a compact metric space and a > 0. We define the following M av m ||· || Ca (X;R) → R Vƒ€C (XR) ||f||₁a = ||f|| + sup {\f(z) - 5(!)); 2,4 € X,₁ x + 1}. f(y)] x, y 0 [d(x, y)]* ||f||∞ = sup{\f(x) : x € X}. BA (a) Show that the function given by (1) is a norm on the space Ca(X;R). BA (b) Show that if a then C(X; R) C Ca(X; R); (c) Show that the space Ca(X; R) equipped with the norm | |, is a Banach space. I 00,0 (1) function by where Let (X, d) b a compact metric space and a > 0. We define the following M av m ||· || Ca (X;R) → R Vƒ€C (XR) ||f||₁a = ||f|| + sup {\f(z) - 5(!)); 2,4 € X,₁ x + 1}. f(y)] x, y 0 [d(x, y)]* ||f||∞ = sup{\f(x) : x € X}. BA (a) Show that the function given by (1) is a norm on the space Ca(X;R). BA (b) Show that if a then C(X; R) C Ca(X; R); (c) Show that the space Ca(X; R) equipped with the norm | |, is a Banach space. I 00,0 (1)
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a To show that the function given by 1 is a norm on the space CX R we need to verify the following properties 1 Nonnegativity For any f CX R Vf 0 This is true because the supremum of nonnegative terms ... View the full answer
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