Interpreting the meaning of the distance between two functions

Question

Given is the function $$f_j = [-2, 1] \rightarrow \mathbb{R} : x \rightarrow x^j, j \in \mathbb{N_0}$$

I'm suppossed to calculate the distance between $f_2$ and $f_1$, or rather $x^2$ and $x$.

Now, given is the distance formula $$<f|g> = \int_a^b{f(x)g(x)dx} | f, g \in C^0([-2,1])$$ and that the distance between two functions can be written as $$d(f, g) = (<f-g|f-g>)^\frac{1}{2}$$ which implies that $$d(f, g) = \sqrt{\int_a^b{(f-g) ^2dx}}$$

If we input our given functions, we find that $$d(f_2, f_1) = \sqrt{\int_{-2} ^1{(x^2-x)^2dx}} = 4.14$$

Now, I'm asking myself how to interpret this result. The two functions cross each other in the origin (which lies within the domain), so how can the distance be $4.14$? Is it the average distance? Or something completly different?

Just trying to make sense of what exactly I'm calculating here and how.

Thank you!

Answer 1

If you want to deal with Riemann integrable functions, then you can approximate the inner product between functions by a Riemann sum. For example, divide the interval $[a,b]$ into $N$ equal intervals of length $\Delta_N=\frac{b-a}{N}$ and sample the functions at the center of each of the $N$ intervals, at say $x_1,x_2,x_3,\cdots,x_N$. Then $$ \langle f,g\rangle_N = \sum_{n=1}^{N}f(x_n)g(x_n)\Delta_N $$ gives a pseudo inner product (meaning positive, but not necessarily positive definite.) And, $$ \lim_{N}\langle f,g\rangle_N = \int_{a}^{b}f(x)g(x)dx. $$ So the inner product is approximated by a discrete inner product where the components of the vectors are point samples of the functions at the intermediate points $x_n$. That puts everything back into the context of inner products of discrete vectors. The components of the vectors are the point-sampled values of the functions. In this way you may think of a function as being approximated by a finite-dimensional vector, and a limit is possible because the total of the weights applied to all components sums to a finite, fixed number--namely to the length of the interval.

Historically, inner products were predated by integral orthogonality conditions between the functions used for Fourier series. The formalism of function "orthogonality" was originally a way to isolate coefficients in expansions of functions into trigonometric functions, and the discovery of orthogonality conditions was an amazing empirical observation that goes back to Euler and Clairaut in the mid 1700's. An abstract connection between such integral expressions and Euclidean norms took over a century to be seen, and then it another 50 years after that before a general inner product was defined in the early 1900's.

The fact that it took so long to see a connection between Euclidean dot products and integral orthogonal conditions should tell you that the connection is strained, and certainly wasn't obvious. So don't be alarmed if the connection seems strained to you, too. Part of the reason for the lack of obvious connection may have been due to the fact that the Riemann integral wasn't defined until 1859. At least we now have the benefit of that hindsight. All integrals were indefinite integrals before that. Riemann created his integral to study Fourier series. Lebesgue continued in this same way when he wrote that he also created his integral to study the convergence of the Fourier series. The evolution to a general notion of dot product or inner product was not straightforward, which is clear from the fact that such an evolution took 150 years. In some sense one is better off finding a connection only through the abstraction of an inner product. That abstraction of inner product was the real breakthrough linking all these ideas.

Answer 2

Two different functions can still have some values in common at some points. The "distance" isn't the minimal distance between points on the graphs, but rather takes into account the distance between function values at all points.

Think of this as you would think of vectors with a finite number of coordinates. Two vectors can have the same first coordinate, but be different vectors; their distance is zero if and only if their coordinates are the same for all coordinates. You can think of functions as generalized vectors that have infinitely many coordinates. Just as the vector $v\in \mathbb R^2$ has coordinates $v_1$ and $v_2$ (which you can think of as $v(1)$ and $v(2)$), the function $f$ has "coordinates" $f_x$ for each $x$ in the domain (we usually write $f(x)$ instead of $f_x$).

I hope this helps.