Gelfand numbers and widths

In general, the Gelfand widths c˜n(T) of a map TT between Banach spaces XX and YY are not equivalent to the Gelfand numbers cn(T)cn(T) of TT. We show that c˜n(T)=cn(T)(n∈N) provided that XX and YY are uniformly convex and uniformly smooth, and TT has trivial kernel and dense range.

► We show that Gelfand numbers and Gelfand widths are not generally equivalent. ► We show conditions under which Gelfand numbers and Gelfand widths are equivalent. ► The distance between polar sets and James orthogonality are discussed.