How to show set of functions is a vector space?

In summary, to show that the set of functions f: S --> k, under function addition and multiplication by a constant is a vector space, one must demonstrate that (f+g)(x) = f(x) + g(x) and that (rf)(x) = rf(x). Additionally, the set of all functions must have the properties of being associative and commutative.

Homework Statement

Let S be a set and V a vector space over the field k. Show that the set of functions f: S --> k, under function addition and multiplication by a constant is a vector space.

Homework Equations

I think I need to show that (f+g)(x) = f(x) + g(x) and that (rf)(x) = rf(x)

The Attempt at a Solution

Are the above conditions true because both functions f and g for this particular case give you k and (k1 + k2) = k1 + k2 and rk = rk. I just don't understand how to show it. Or do I just literally write (f+g)(x) = f(x) + g(x) and that (rf)(x) = rf(x).

Yes, that's exactly right. The set of all functions, with the standard definitions of "sum of two functions" and "product of a function and a number", is a vector space because for any two functions f and g, f+ g is also a function and for any function f and real number r, rf is a function.

Strictly speaking you should also show or at least state that addition of functions is associative, commutative, etc. but those might be assumed.

1. What are the requirements for a set of functions to be considered a vector space?

In order for a set of functions to be considered a vector space, it must satisfy the following requirements:

• Addition closure: The sum of two functions in the set must also be in the set.
• Multiplication closure: The scalar product of a function and a scalar must also be in the set.
• Commutativity: The order of addition does not affect the result.
• Associativity: The grouping of operations does not affect the result.
• Existence of a zero vector: There must be a function in the set that acts as the additive identity.
• Existence of additive inverses: For every function in the set, there must be another function in the set that when added together, results in the zero vector.
• Distributivity: The scalar product of a sum of functions must be equal to the sum of scalar products of individual functions.

2. How do I prove that a set of functions is a vector space?

To prove that a set of functions is a vector space, you must show that it satisfies all of the requirements listed in the answer to the previous question. This can be done by providing specific examples and using mathematical proofs to demonstrate that the set of functions follows the properties of a vector space.

3. Can a set of functions be a vector space if it contains only one function?

Yes, a set of functions can be a vector space if it contains only one function. This function must satisfy all of the requirements for a vector space, such as being closed under addition and multiplication, having a zero vector, and having additive inverses.

4. How does the dimension of a vector space relate to the number of functions in the set?

The dimension of a vector space is the number of linearly independent functions in the set. This means that the number of functions in the set does not necessarily determine the dimension of the vector space. For example, a set of three functions could have a dimension of two if one of the functions can be expressed as a linear combination of the other two.

5. Can a set of functions be a vector space if it contains non-continuous functions?

Yes, a set of functions can still be a vector space even if it contains non-continuous functions. The requirements for a vector space do not specify that the functions must be continuous. However, if the set of functions is being used to represent a physical system, it may be more useful to have continuous functions in order to accurately model the system.

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