Math 6210 • Real Analysis

Let \(X_1,X_2\) be Banach spaces, \(T\) a linear map of \(X_1\) into \(X_2\), and \(\{\lambda _\alpha \}_{\alpha \in A}\subset X_2^*\) be a subset which separates points of \(X_2\); that is, if \(v_1,v_2\in X_2\), \(v_1\ne v_2\), then there exists \(\alpha \in A\) such that \(\lambda _\alpha (v_1)\ne \lambda _\alpha (v_2)\). Suppose that \(\lambda _\alpha \circ T\in X_1^*\) for all \(\alpha \in A\). Then show that \(T\) is continuous.