What is the subspace spanned by a single vector in function space?

[cahiersujet]

What would the subspace spanned by a single vector (for example) f(x)=x+1 be?

 

[yyat]

The subspace spanned by a vector is the set of all scalar multiples of that vector. A function like f(x)=x+1 is a "vector" in some function space, most likely over the real or complex numbers, so the answer is all functions of the form g(x)=a*(x+1), where a is an element of the scalar field (R or C).