Let C[a, b] denote the set of all real-valued functions that are continuous on the interval...

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Let C[a, b] denote the set of all real-valued functions that are continuous on the interval [a, b] and use the standard function addition and scalar multiplication for these functions. That is for, f(x), g(x) = C[a, b] and real number c, we define the sum of two functions and scalar multiplication of a function as: (f + g)(x) = f(x) + g(x) (cf)(x) = cf(x) Show that C[a, b] with the given operations is a vector space (Show that all properties are satisfied) Let C[a, b] denote the set of all real-valued functions that are continuous on the interval [a, b] and use the standard function addition and scalar multiplication for these functions. That is for, f(x), g(x) = C[a, b] and real number c, we define the sum of two functions and scalar multiplication of a function as: (f + g)(x) = f(x) + g(x) (cf)(x) = cf(x) Show that C[a, b] with the given operations is a vector space (Show that all properties are satisfied)