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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

2003 • Question 5

Question

Math 6210 • Real Analysis 2003 • Question 5

Let \(A\) be a closed subspace of \(C([0,1])\) which is also closed in \(L^2([0,1])\). Prove that the orthogonal projection \(P\) which projects \(L^2([0,1])\) onto \(A\) is continuous as a linear transformation from \(L^2([0,1])\) into \(C([0,1])\). Note that \(C([0,1])\) has the topology determined by the norm \(\|\cdot \|_\infty \), while \(L^2([0,1])\) has the topology determined by the norm \(\|\cdot \|_2\).

Raw LaTeX

Let \(A\) be a closed subspace of \(C([0,1])\) which is also closed in
\(L^2([0,1])\). Prove that the orthogonal projection \(P\) which projects
\(L^2([0,1])\) onto \(A\) is continuous as a linear transformation from
\(L^2([0,1])\) into \(C([0,1])\). Note that \(C([0,1])\) has the topology determined
by the norm \(\|\cdot\|_\infty\), while \(L^2([0,1])\) has the topology determined
by the norm \(\|\cdot\|_2\).

Topic Tags

Hilbert Spaces and Orthogonality Lp Spaces and Sequence Spaces Weak Topologies and Weak Convergence