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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Summer 2018 • Question 2

Question

Math 6210 • Real Analysis Summer 2018 • Question 2

Let \(V\) be a normed space and \(U\) a closed subspace. For every \(x\in V\), let \[ \lVert x+U\rVert =\inf \{\lVert x+y\rVert :\ y\in U\}. \]

(a) Prove that \(\lVert x+U\rVert \) is a norm on \(V/U\).

(b) If \(V\) is complete, so is \(V/U\).

Raw LaTeX

Let \(V\) be a normed space and \(U\) a closed subspace. For every
\(x\in V\), let
\[
\lVert x+U\rVert=\inf\{\lVert x+y\rVert:\ y\in U\}.
\]

(a) Prove that \(\lVert x+U\rVert\) is a norm on \(V/U\).

(b) If \(V\) is complete, so is \(V/U\).

Topic Tags

Banach Spaces Hilbert Spaces and Orthogonality