On automatic continuity and three problems of ``The Scottish Book'' concerning the boundedness of polynomial functionals
Let \(X\) and \(Y\) be normed linear spaces. A polynomial operator of degree \(m\), \(P:X\to Y\), is a mapping defined by \(x\mapsto P_0(x)+\dots+ P_m(x)\), where \(P_k\) is the restriction to the \(k\)-fold diagonal of a \(k\)-linear operator from \(X^k\) into \(Y\) for each \(k\in \{1,\dots, m\}\) and \(P_0(x)\) is a constant. \(P\) is called a polynomial functional if \(Y\) equals the real or complex numbers. Investigating questions of automatic continuity it is shown that for discontinuous polynomial operators there exists a sequence \((z_i)\) in \(X\) converging to 0 such that \(\sup\| P(x_0+z_i)\|= \infty\) for each \(x_0\in X\). Further, if the linear dimension of \(X\) is \(\omega_1\) there exists a discontinuous polynomial functional \(p\) of degree 2 such that no sequence \((x_i)\) in \(X\) converging to 0 has the property \(p(x+x_i)\to \infty\) for each \(x\in X\). As a weakening of the notion continuity a mapping \(F:X\to Y\) is called isotropic iff it is continuous or there exists a sequence \((x_i)\) in \(X\) converging to 0 and a constant \(c>0\) such that \(\varlimsup \text{dist} (F(x+x_i), F(x))\geq c\) for each \(x\in X\). Theorems of the closed graph type are presented such as: If \(X\) is a complete metric group and \(Y\) is a separable metric space, \(F\) is continuous if it is isotropic and has a Borel graph. Finally, the question of the existence of infinite-dimensional affine subspaces in the kernel of a polynomial functional is treated.
scientific article; zbMATH DE number 1218574
scientific article