What is the difference between the 'Big O' set and a span of the vector space for a function?

[I'm assuming you meant to say "...then f(x) ∈ Span(g(x))".]

It's true if g(x) is a scalar multiple of f(x), but not if we add constants.

For example, consider f(x) = 2x + 3 and g(x) = 2x + 5.

Certainly, f(x) ∈ O(g(x)), since both 2x + 3 ∈ Ɵ(x) and 2x + 5 ∈ Ɵ(x).

But f(x) is not in the span of g(x), since f and g are not scalar multiples of each other.

in case you have a collection of linearly self sufficient vectors V and a vector area S, such that each and every vector in S may be shaped as a linear mix of vectors in V, then S is the span of V, and V is a spanning set of S. that's basically the two names for the two gadgets in this dating. a collection of vectors ("spanning set") is asserted to "span" the set S (span used as a verb). yet S, the gap which V spans, is stated as the "span" (noun) of V.

They aren't that related.

f(x) ∈ O(g(x)) if and only if lim(x-->∞) |f(x)/g(x)| < ∞

f(x) ∈ Span(f(x)) is always true because there exists a real number k such that k*f(x) = f(x) namely k=1. (I assume it is vector space over the real numbers).

f(x) ∈ Span(g(x)) is generally false. For example x ∈ O(x^2) but there is no real number k such that k*x^2 = x. Hence x ∉ Span(x^2)