August 12th, 2015, 12:47 PM  #1 
Member Joined: Mar 2015 From: Brasil Posts: 90 Thanks: 4  Vector Spaces
Help!!

August 12th, 2015, 01:05 PM  #2 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,807 Thanks: 1045 Math Focus: Elementary mathematics and beyond 
Can a translation be provided?

August 13th, 2015, 07:39 AM  #3 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,168 Thanks: 867 
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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