The Approximation Property without the Bounded Approximation Property in a Separable Banach Space and Its Dual
We investigate whether a dual Banach space with the approximation property (AP) must necessarily have the bounded approximation property (BAP). Using a quantitative renorming theorem for the space of absolutely summable sequences, we establish a direct-sum construction yielding a negative answer. More precisely, we construct a separable Banach space (X) such that both (X) and its dual (X^*) have the approximation property, while neither space has the bounded approximation property. The construction uses a sequence of equivalent renormings of (\ell_1) with increasingly restrictive finite-dimensional approximation bounds. By assembling these spaces through a (c_0)-direct sum, we obtain a dual space whose approximation property is preserved, but whose approximation constants cannot be uniformly bounded. We establish the required transfer principles, including the preservation of approximation bounds under complemented coordinate projections and the descent of the bounded approximation property from a dual space to its predual. We also provide an explicit complex-scalar version of the construction. The argument separates the external quantitative renorming input from the direct-sum construction and its functional-analytic consequences. In particular, the resulting example demonstrates that the approximation property of a dual space does not imply its bounded approximation property, even when the predual is separable and both spaces possess the approximation property.