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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2018 • Question 3

Question

Math 6210 • Real Analysis Spring 2018 • Question 3

Let \(T:V\to W\) be a bounded linear map, where \(V\) and \(W\) are normed spaces. Let \(U\) be the kernel of \(T\). Prove that \(\lVert S\rVert =\lVert T\rVert \) where \(S:V/U\to W\) is such that \[ S(x+U)=T(x) \] for all \(x\in V\).

Raw LaTeX

Let \(T:V\to W\) be a bounded linear map, where \(V\) and \(W\) are
normed spaces. Let \(U\) be the kernel of \(T\). Prove that \(\lVert S\rVert
=\lVert T\rVert\) where \(S:V/U\to W\) is such that
\[
S(x+U)=T(x)
\]
for all \(x\in V\).

Topic Tags

Banach Spaces Bounded Linear Operators