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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 1995 • Question 3

Question

Math 6210 • Real Analysis Fall 1995 • Question 3

Let \(B\) be a Banach space and \(A\subset B\) a closed subspace. If \(x\in B\) but \(x\notin A\), prove that there is a bounded linear functional \(f\) on \(B\) such that \(f(x)=1\), \(f(y)=0\) for every \(y\in A\) and \(\lVert f\rVert =(\inf \{\lVert x+y\rVert :y\in A\})^{-1}\).

Raw LaTeX

Let \(B\) be a Banach space and \(A\subset B\) a closed subspace. If
\(x\in B\) but \(x\notin A\), prove that there is a bounded linear functional \(f\)
on \(B\) such that \(f(x)=1\), \(f(y)=0\) for every \(y\in A\) and
\(\lVert f\rVert=(\inf\{\lVert x+y\rVert:y\in A\})^{-1}\).

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Topic Tags

Banach Spaces Bounded Linear Operators Dual Spaces and Hahn-Banach Theorem Hilbert Spaces and Orthogonality