I know that Byzantium is actually just the Eastern Roman Empire and the true continuation of Rome, but what did the other European kingdoms think of it? It would seem that the HRE would argue that they were the real Roman Empire. Did European states recognize that Byzantium was the Roman Empire? If not, what did they address it as?
Some of them definitely did. For example, in 507 the Frankish king Clovis accepted the title of Consul from the Byzantine emporer, which means he technically became a vasal of Constantinople (although he remained de facto independent). That fact that he accepted this shows that the emporer still had a lot of prestige in Western Europe. When Charlemange had himself declared Roman Emporer in 800, the Byzantines saw it as an usurpation of their power and even launched a war against the Frankish Empire.
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I got a few answers regarding the early middle ages, but what did Europeans think of Byzantium during the high and late middle ages? Were they still a force to be reckoned with? Were they still seen as the continuation of Rome?