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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

2006 • Question 4

Question

Math 6210 • Real Analysis 2006 • Question 4

Let \(E_1,\ldots ,E_n\) by a finite collection of disjoint measurable subsets of \(X\) and let \(M\) be a subspace of \(L^2(X)\) spanned by the functions \(\{u_1,\ldots ,u_n\}\) where each \(u_i\) is in \(L^2(X)\) and has support contained in \(E_i\). Let \(P:L^2(X)\to \mathbb R\) be the orthogonal projection. Find a function \(h:X\times X\to \mathbb R\) such that \[ Pf(x)=\int _X h(x,y)f(y)\,d\mu (y) \] for all \(f\in L^2(X)\).

Raw LaTeX

Let \(E_1,\ldots,E_n\) by a finite collection of disjoint measurable
subsets of \(X\) and let \(M\) be a subspace of \(L^2(X)\) spanned by the functions
\(\{u_1,\ldots,u_n\}\) where each \(u_i\) is in \(L^2(X)\) and has support contained in
\(E_i\). Let \(P:L^2(X)\to\mathbb R\) be the orthogonal projection. Find a function
\(h:X\times X\to\mathbb R\) such that
\[
Pf(x)=\int_X h(x,y)f(y)\,d\mu(y)
\]
for all \(f\in L^2(X)\).

Topic Tags

Hilbert Spaces and Orthogonality Lebesgue Integration Lp Spaces and Sequence Spaces Measurable Functions