Question 1.11.3: From calculus, consider the set C[−1,1] of all continuous fu...

Differential Equations with Linear Algebra [EXP-50954]

From calculus, consider the set C[−1,1] of all continuous functions on the interval [−1,1]. That is,

C[−1,1] = {f | f is continuous on [−1,1]}.

Show that C[−1,1] is closed under addition and scalar multiplication, and also that C[−1,1] contains a zero element.

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Two standard facts from calculus tell us that the sum of any two continuous functions is also a continuous function and that a constant multiple of a continuous function is also a continuous function. Thus C[−1,1] is closed under addition and scalar multiplication. Furthermore, the zero function z(x)=0 is itself continuous, which shows that C[−1,1] indeed has a zero element.

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