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Exercise 6 6 4 Let X be any set and consider the set of functions...

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Exercise 6 6 4 Let X be any set and consider the set of functions S f X R Notice we can add two functions f g x f x g x we can scale a function a f x a f x and we have a zero function z x 0 Prove two properties to start to see that S is a vector space Now from here 5 a Look at some Calculus textbook tell me which book you used the library has some and the internet has tons and tell me where you found the properties which would help you see why the set of continuous functions with the same domain is a vector space b Now assuming continuous functions are a vector space prove to me that differentiable functions are a subspace using the definition of subspace and some properties from Calculus c Now assuming differentiable functions are a vector space prove to me that polynomial functions are a subspace of differentiable functions using the definition of subspace and properties from Algebra

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