A non-split short complex of Banach spaces whose dual splits.

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0 ⟶ c_{0} ⟶^{i} l^{∞} ⟶^{π} \frac{l^{∞}}{c_{0}} ⟶0 is a short exact complex of Banach spaces which doesn’t split since c_{0} is not complemented in l^{∞}. Its dual complex 0 ⟶ (\frac{l^{∞}}{c_{0}} )^{#} ⟶^{π^{#}} (l^{∞})^{#} = (l^{1})^{##} ⟶^{i^{#}} c_{0}^{#} = l^{1}⟶0 splits. Notice that the later complex is exact and the canonical embedding l^{1} ⟶ (l^{1})^{##} is a right inverse to i^{#}.