In this question someone asks about showing that the set of all functions of the form $y(t) = c_1\cos\omega t + c_2\sin\omega t$ is a vector space. But doesn't literally any set of functions of the form $y(t) = c_1f(t) + c_2g(t) + \ldots$ form a vector space? After all, there will always be a zero element (coefficients = 0) and an additive inverse (coefficients of opposite sign), and trivially scaling or adding two $y(t)$ will yield another function of the same form.
So what is an example of a set of functions that do NOT form a vector space? The most common pedagogic example I've seen is unsatisfyingly contrived: the set of all polynomials of degree N. This is explained to not form a vector space because the zero element is not of degree N. However technically the zero element is still of the form $c_1 + c_2x + \ldots + c_Nx^N$, so I'm not sure how comfortable I am with this example.