Show that C'[a, b] is subspace of C [a, b]

Kaycee Roche 2020-12-30 Answered
Show that C'[a, b] is subspace of C [a, b]
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Expert Answer

Cristiano Sears
Answered 2020-12-31 Author has 96 answers
To shiw that C'[a.b] is subspace of C[a,b]. Suppose f1 and f2 are in C'[a.b]. We will prove that f1+f2 is in C'[a,b]. Let f=f1+f2. Then from the definition of derivative we have
limh0f(x+h)f(x)h=limh0(f1(x+h)f1(x)h)+f2(x+h)f2(x)h))=(limh0(f1(x+h)f1(x)h))+limh0(f2(x+h)f2(x)h)).
Whis last equality is true if both of the limits on the last line exist, and it is given that they do. Saying the lat equality is true means the limits involved exist. The relevant theorem is that if two limits as h something equal to the sum of those two limits.
The first equality above is just algebra: adding factions. Nowto show for cR, cf1C[a,b]. We know that
limh0cf1(x+h)cf(x)h=climh0f1(x+h)f1(x)h
By similar argument, since limit of right side part of above equation exists so, cf1 is also differentiable, therefore, cf1C[a,b]. Hence C'[a,b] is a subspace of C[a,b].

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