The fundamental-limit theory of antennas provides a theoretical limit to assist in the evaluation of antenna performance in terms of antenna size, fractional impedance bandwidth, and gain. The limit is very useful in practice, giving a basis for restricting the design search to a class of realizable antennas based on size and performance. Previous research on the limit theory focused on electrically small, resonant antennas. In this paper, we discuss how the classical fundamental-limit theory can be interpreted for ultrawideband antennas. The frequency response of Chu's equivalent circuit model for spherical modes suggests the concept of an ideal antenna. The transfer function of the ideal antenna, showing ultrawideband antenna characteristics, simply has an entire function and two complex poles. In this paper, an antenna design strategy is developed based on observations of ideal antenna characteristics. A process is presented for evaluating how well an ultrawideband antenna can approach the theoretical size limit based on the 3-dB cutoff frequencies of spherical modes.