Which of the following subsets are subspaces of C[0,1]? Justify
i) { f C[0,1] | f(0) = f(1)}
ii){ f C[0,1]| f(0) = 0 and f(1) = 0}
iii){ f C[0,1]|f(0) = 0 or f(1)=0}
iv){ f C[0,1]| f is nondecreasing }
C[0, 1] is the set of functions that are continuous on the interval [0, 1].
i) If f and g are two functions, such that f(0)= f(1) and g(0)= g(1), is that also true for af(x)+ bg(x) where a and b can be any real numbers?
ii) If f and g are two functions, such that f(0)= f(1)= 0 and g(0)= g(1)= 0, is that also true for af(x)+ bg(x) where a and b can be any real numbers?
iii) If f and g are two functions, such that either f(0)= 0 or f(1)= 0 and either g(0)= 0 or g(1)= 0, is that also true for af(x)+ bg(x) where a and b can be any real numbers? (Be careful here. What if f(0)= 0 but and g(1)= 0 but ?)
iv) If f and g are two nondecreasing functions, is af(x)+ bg(x) also nondecreasing, where a and b can be any real numbers? (Think about a and b being negative.)