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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2013 • Question 2

Question

Math 6210 • Real Analysis Fall 2013 • Question 2

Let \(V\) and \(W\) be Banach spaces, and \(L(V,W)\) the space of bounded linear maps from \(V\) to \(W\). Suppose that \(T_n\in L(V,W)\), \(n=1,2,\ldots \), and that \(\lim T_nx\) exists for each \(x\in V\). Show that \(T:V\to W\) defined by \(Tx=\lim T_nx\) is in \(L(V,W)\).

Raw LaTeX

Let \(V\) and \(W\) be Banach spaces, and \(L(V,W)\) the space of
bounded linear maps from \(V\) to \(W\). Suppose that \(T_n\in L(V,W)\),
\(n=1,2,\ldots\), and that \(\lim T_nx\) exists for each \(x\in V\). Show that
\(T:V\to W\) defined by \(Tx=\lim T_nx\) is in \(L(V,W)\).

Topic Tags

Banach Spaces Bounded Linear Operators