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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2010 • Question 3

Question

Math 6210 • Real Analysis Spring 2010 • Question 3

Let \(H\) be a Hilbert space, and let \(V\subsetneq H\) be a nonzero proper closed subspace. Let \(\pi _V\) be the orthogonal projection onto \(V\).

(a) Show that \(\pi _V\) has norm \(1\), is idempotent (\(\pi _V^2=\pi _V\)), and is self-adjoint (\(\pi _V^*=\pi _V\)).

(b) Conversely, if \(P:H\to H\) is a self-adjoint, idempotent, continuous linear transformation of operator norm \(1\), show that \(P=\pi _V\) for some closed subspace \(V\) of \(H\).

Raw LaTeX

Let \(H\) be a Hilbert space, and let \(V\subsetneq H\) be a nonzero
proper closed subspace. Let \(\pi_V\) be the orthogonal projection onto \(V\).

(a) Show that \(\pi_V\) has norm \(1\), is idempotent
(\(\pi_V^2=\pi_V\)), and is self-adjoint (\(\pi_V^*=\pi_V\)).

(b) Conversely, if \(P:H\to H\) is a self-adjoint, idempotent, continuous
linear transformation of operator norm \(1\), show that \(P=\pi_V\) for some
closed subspace \(V\) of \(H\).

Topic Tags

Hilbert Spaces and Orthogonality Zeros of Holomorphic Functions and the Identity Theorem