NOVELTY - A polymer has pendant group(s) comprising carboxyl group or its salt, and group interposed between principal chain and carboxyl group or its salt. The group interposed between principle chain and carboxyl group or its salt is hydrocarbon group or perfluorocarbon group, hydrocarbon group and ester group and/or carbonate group, perfluorocarbon group and ester group and/or carbonate group, and carbonyl carbon in the carboxyl group or its salt is directly bonded to carbon atom in the hydrocarbon group or the perfluorocarbon group.    USE - Polymer is used in secondary battery (claimed) e.g. alkali electrolyte solution secondary battery, lithium ion secondary battery, solid secondary battery. Uses include but are not limited to electrical double layer capacitor, dye-sensitized solar cell, dispersing agent, solubilizer, surface-control agent, hydrogel substitute material and oiling agent.    ADVANTAGE - The polymer has excellent oxidation resistance and ionic dissociation property in organic solvent, and exhibits functions based on a carboxyl group or its salt by coexisting with organic solvent. The secondary battery obtained using polymer has excellent charging/discharging cycle characteristics, durability and high-temperature-storage property, and is safe.    DETAILED DESCRIPTION - A polymer has pendant group(s) comprising carboxyl group or its salt, and group interposed between principal chain and carboxyl group or its salt. The group interposed between principle chain and carboxyl group or its salt is hydrocarbon group or perfluorocarbon group, hydrocarbon group and ester group and/or carbonate group, perfluorocarbon group and ester group and/or carbonate group, and carbonyl carbon in the carboxyl group or its salt is directly bonded to carbon atom in the hydrocarbon group or the perfluorocarbon group. When group interposed between principle chain and carboxyl group or its salt is hydrocarbon group or hydrocarbon group, and ester group and/or carbonate group. The fluorine is bonded to carbon of hydrocarbon at ( alpha )-position or ( beta )-position with respect to carbonyl carbon of carboxyl group or its salt. An INDEPENDENT CLAIM is included for secondary battery, which consists of positive electrode, negative electrode, separator and electrolyte containing positive electrode active material comprising polymer.