Norms, inner products and contractions
Different norms of vectors, lengths and inner products of functions through integrals, and contraction maps with their fixed points.
How to solve it
A norm is a way of measuring length. A vector has several: the sum of the absolute coordinates ($\ell_1$), the usual length ($\ell_2$), the largest absolute coordinate ($\ell_\infty$). Functions can be measured and added like vectors too: the inner product is the integral of the product, and sines and cosines of different frequencies are perpendicular.
Step by step
- $\|x\|_1$ is the sum of the absolute values, $\|x\|_2$ the root of the sum of squares, $\|x\|_\infty$ the largest absolute value.
- The inner product of functions on $[-\pi,\ \pi]$: $\langle f, g\rangle = \int f g\,dx$. Products of different harmonics give zero, and $\int \cos^2 nx\,dx = \int \sin^2 nx\,dx = \pi$.
- The length of a trigonometric polynomial is a Pythagorean theorem: $\|f\|^2 = \pi \cdot (\text{the sum of the squared coefficients})$.
- A contraction $T$ shrinks distances by a factor $q < 1$. Its fixed point is unique: solve $T(x) = x$.
Common mistakes
- Adding the coordinates with their signs in $\|x\|_1$ instead of their absolute values.
- Expanding the square for a function's length and computing every cross integral — they vanish by orthogonality.
- Forgetting the factor $\pi$: $\int_{-\pi}^{\pi} \cos^2 x\,dx = \pi$, not $1$.
- Calling a map with $|q| \ge 1$ a contraction.