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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2019 • Question 2

Question

Math 6210 • Real Analysis Fall 2019 • Question 2

Let \(V\) be a normed space and \(U\) a closed subspace. For every \(x\in V\), let \[ \lVert x+U\rVert =\inf \{\lVert x+y\rVert :\ y\in U\}. \] Then \(\lVert x+U\rVert \) is a norm on \(V/U\). (You do not need to prove that.) Assume \(U\ne V\). Prove that the map \(P:V\to V/U\) defined by \(P(x)=x+U\) has operator norm \(1\).

Raw LaTeX

Let \(V\) be a normed space and \(U\) a closed subspace. For every
\(x\in V\), let
\[
\lVert x+U\rVert=\inf\{\lVert x+y\rVert:\ y\in U\}.
\]
Then \(\lVert x+U\rVert\) is a norm on \(V/U\). (You do not need to prove that.)
Assume \(U\ne V\). Prove that the map \(P:V\to V/U\) defined by \(P(x)=x+U\) has
operator norm \(1\).

Topic Tags

Banach Spaces Hilbert Spaces and Orthogonality