MathDB
IMC 2008 Day 2 P6 - Hilbert space

Source: Problem 6

July 28, 2008
vectorFunctional AnalysisIMCcollege contests

Problem Statement

Let H \mathcal{H} be an infinite-dimensional Hilbert space, let d>0 d>0, and suppose that S S is a set of points (not necessarily countable) in H \mathcal{H} such that the distance between any two distinct points in S S is equal to d d. Show that there is a point yāˆˆH y\in\mathcal{H} such that \left\{\frac{\sqrt{2}}{d}(x\minus{}y): \ x\in S\right\} is an orthonormal system of vectors in H \mathcal{H}.
Back to ProblemsView on AoPS