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The establishment of Constantinople did not weaken the western part of the empire. Constantinople was made the capital of the eastern part of the empire. Milan was made the capital of the western part. Rome became the nominal capital of the whole empire.
Making Milan the capital of the west was designed to strengthen the western part. Milan was nearer the front of the frontiers along the Alps and the river Rhine which were under constant attacks. It was also where the cavalry-only forces, established by Emperor Gallienus as a rapid deployment force to reach the areas of the frontier under attack quickly, were based.