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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 1994 • Question 4

Question

Math 6210 • Real Analysis Fall 1994 • Question 4

Let \(A\), \(B\), and \(C\) be Banach spaces and let \(f:A\to C\) and \(g:B\to C\) be bounded linear maps. Suppose that \(g\) is one to one and \(f(A)\subset g(B)\). Prove that there is a bounded linear map \(h:A\to B\) such that \(b=h(a)\) if and only if \(f(a)=g(b)\).

Raw LaTeX

Let \(A\), \(B\), and \(C\) be Banach spaces and let \(f:A\to C\) and
\(g:B\to C\) be bounded linear maps. Suppose that \(g\) is one to one and
\(f(A)\subset g(B)\). Prove that there is a bounded linear map \(h:A\to B\) such
that \(b=h(a)\) if and only if \(f(a)=g(b)\).

Topic Tags

Banach Spaces Bounded Linear Operators