ISEGORIA / MATH ENCYCLOPEDIA
Functional analysis: norms, projections, and operators
Compare norms, orthogonal projections, and the spectrum of a small operator.
Before you begin: Linear algebra and real analysis
Predict, manipulate, then check your reasoning against the example and question. Graphs illustrate the mathematics; they do not replace a proof.
1. Norms and unit balls
Change p and watch the unit ball move from the p = 1 diamond through the p = 2 circle towards the p = ∞ square. Drag the vector x: its p-norm is how far the ball must be scaled to reach it, and the right panel shows that this number decreases as p grows. Below p = 1 the ball stops being convex.
Worked example. For \(p=2\), the unit ball is the Euclidean circle.
Watch out. For p < 1 the formula still defines a unit ball, but the triangle inequality fails, so it is not a norm. The p = ∞ square is drawn exactly as the maximum norm.
Which unit ball has the sharpest corners?
The \(p=1\) diamond.
2. Orthogonal projection
Drag the vector v and the direction of the line. The projection is the point of the line closest to v, which is where the smallest ball around v touches the line. With the weight w = 1 the inner product is the dot product and the residual looks perpendicular; raise w and the balls become ellipses, so the projection moves although the picture of v and the line has not changed.
Worked example. The residual is orthogonal to the subspace: \(\langle v-\operatorname{proj}_u v,u\rangle=0\).
Watch out. Projection depends on the chosen inner product, not only on the drawn direction.
What vanishes at the projection?
The inner product of the residual with the direction.
3. Operator spectrum
The operator A has eigenvalues λ₁ and λ₂ along two eigenvectors 60° apart. Repeated action on a vector you drag stretches it by |λ| along each eigendirection, so Aⁿx and the image of the unit circle turn towards the eigenline whose eigenvalue is largest in size. The right panel shows the smallest singular value of A − λI, which vanishes exactly at the eigenvalues: that is where A − λI stops being invertible.
Worked example. With \(\lambda_1=1.2\), \(\lambda_2=0.5\) and eigenvectors \(e_1=(1,0)\), \(e_2=(\cos 60^\circ,\sin 60^\circ)\), write \(x=a e_1+b e_2\). Then \(A^n x=1.2^n a\,e_1+0.5^n b\,e_2\): after four steps the \(e_2\) part is down to \(1/16\) of its size while the \(e_1\) part has grown by \(1.2^4\approx 2.07\), so \(A^4x\) lies close to the \(e_1\) line.
Watch out. A finite matrix spectrum is a teaching proxy for the richer spectra of infinite-dimensional operators.
What happens when an eigenvalue crosses zero?
The operator loses invertibility along that eigendirection.