Could someone help me solving this example?

[demmimur]

Let V be the set of all the functions of a non-empty set X in a body k. For any functions f, g ∈ V and any scalar k ∈ K, let f + g and kf be the functions in V
defined as follows:
(f + g) (x) = f (x) + g (x) and (kf) (x) = kf (x), ∀ x ∈ X. Show that V is a vector space over K

 

[Steven G]

Sure we can help you solving your example. That is never a problem on this forum.

What do you need help with? Where are you stuck?Can we please see the work you did with this problem. Please give us something to work with so we can help you.

 

[Dr.Peterson]

Let V be the set of all the functions of a non-empty set X in a body k. For any functions f, g ∈ V and any scalar k ∈ K, let f + g and kf be the functions in V
defined as follows:
(f + g) (x) = f (x) + g (x) and (kf) (x) = kf (x), ∀ x ∈ X. Show that V is a vector space over K

I think the word "body" was translated from some other language into the wrong English word; and I think you meant K, not k. Is K perhaps a field? Or something else?

In any case, assuming that you have learned the definition of a vector space, can you at least state what that definition says, when applied to your set V? That's the first step in a proof like this.

 

[pka]

Let V be the set of all the functions of a non-empty set X in a body k. For any functions f, g ∈ V and any scalar k ∈ K, let f + g and kf be the functions in V defined as follows: (f + g) (x) = f (x) + g (x) and (kf) (x) = kf (x), ∀ x ∈ X. Show that V is a vector space over K

There is a foundational problem with your question.
Almost any textbook for Linear Analysis (vector spaces or linear spaces) gives what you posted as the definition for a vector space.
One does not prove a definition. So you need to tell us from where this question comes, what are the definition to be used, what other definitions are you given.

 

[Dr.Peterson]

I wouldn't say that what is provided is merely the definition of a vector space. It is a definition of the operations on functions, which are then to be proved to form a vector space.

It is, however, so similar to some of the axioms for a vector space that the proof requested will be quite simple. This is perfectly fitting for a first introduction to the concept.