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August 12th, 2015, 12:47 PM   #1
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Vector Spaces

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August 12th, 2015, 01:05 PM   #2
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Can a translation be provided?
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August 13th, 2015, 07:39 AM   #3
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I don't pretend to speak English, but I think it says:

For the set of all real valued functions, defined on set S, we define addition by [f+ g](x)= f(x)+ g(x) for all x and y in S, and scalar multiplication by (af)(x)= af(x) for all x in S and a any real number.

Prove that this set, with these operations, satisfies all the requirements of the definition of "Vector Space".

Those requirements are:
The set is closed under addition: if f and g are such functions, the f+ g is also a function from S to real numbers.
The set is close under multiplication: if f is such a function and a is a real number, then af is a function from S to real numbers.
One also needs to show that addition is commutative and associative, scalar multiplication is associative in the sense that a(bf)= (ab)f, that there exist an additive identitity, and each function has an additive inverse.

Those follow almost immediately from the fact that, for any x, f(x) is a real number and those properties are true of real numbers.
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