F10HW_9 - F2010. Homework # 9. Section 4.2 7 a) The set of...

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F2010. Homework # 9. Section 4.2 7 a) The set of continuous real valued functions on R is not empty. It contains at least the function constant equal to 0. The sum of two continuous functions is continuous. If we consider the function constant and equal to c , this function is continuous. If f is any continuous function from R to R , the product cf is continuous as product of continuous functions. The set of continuous real valued functions on R is not empty, closed for addition and scalar multiplication, therefore it is a subspace. b) Exactly as a) by replacing continuous by differentiable . c) The set of real valued functions on the interval [0 , 1] such that f (1 / 2) = 0 is not empty. It contains at least the function constant equal to 0. If f and g are real valued functions on the interval [0 , 1] such that f (1 / 2) = 0 and g (1 / 2) = 0, their sum is a real valued functions on the interval [0 , 1], and ( f + g )(1 / 2) = f (1 / 2) + g (1 / 2) = 0. If f is a real valued functions on the interval [0 , 1] such that f (1 / 2) = 0 and c is a real number, cf is a real valued functions on the interval [0 , 1], and ( cf )(1 / 2) = cf (1 / 2) = 0. The set of real valued functions f on the interval [0 , 1] such that f (1 / 2) = 0 is not empty, it is closed for addition and scalar multiplication, therefore it is a subspace. Remark. The set of real valued functions on the interval [0 , 1] such that f (1 / 2) = a, a ̸ = 0 is not a subspace. (0 does not belong to it, it is not closed for addition or scalar multiplication.) d) The set of real valued functions on the interval [0 , 1] such that 1 0 f ( x ) dx = 0 is not empty. It contains at least the function constant equal to 0. If 1 0 f ( x ) dx = 0 and 1 0 g ( x ) dx = 0, then 1 0 ( f + g )( x ) dx = 1 0 f ( x ) dx + 1 0 g ( x ) dx = 0. So closed for addition. If
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F10HW_9 - F2010. Homework # 9. Section 4.2 7 a) The set of...

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