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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6710 • Applied Linear Operators and Spectral Methods

Fall 2023 • Question 5

Question

Math 6710 • Applied Linear Operators and Spectral Methods Fall 2023 • Question 5

Let \(X\) be a Banach space. Let \(y \in X\) be fixed and let \(f : X \to \mathbb {R}\) be a bounded linear functional. Show that there is a constant \(C > 0\) such that for all scalars \(\alpha \) with \(|\alpha | < C\) the equation

\[x + \alpha f(x)x = y\]

has a unique solution on the open ball \(\{x \in X : \|x - y\| < 1\}\).

Raw LaTeX

Let \(X\) be a Banach space. Let \(y \in X\) be fixed and let \(f : X \to \mathbb{R}\) be a bounded linear
functional. Show that there is a constant \(C > 0\) such that for all scalars \(\alpha\) with \(|\alpha| < C\)
the equation

\[x + \alpha f(x)x = y\]

has a unique solution on the open ball \(\{x \in X : \|x - y\| < 1\}\).

Topic Tags

Banach and Normed Spaces Bounded Linear Operators Dual Spaces and Bounded Linear Functionals