Analogy to Vectors in L2-Space

Question

I am trying to visualize the L2-space as a flat space with points as functions. What is the analogous idea of vectors in Euclidean space? My intuition is the "vector" between two functions would be the linear interpolation between them. Perhaps geodesic is the proper term here? An example, restricting our domain to $(0,1)$ with $f = \frac{1}{x^2}$ and $g = \frac{1}{x^3}$, then $\lambda f + (1-\lambda)g$ with $0\leq\lambda\leq1$ would draw the "vector" from $f$ to $g$.