Evidence for who Robin Hood is
The first evidence for the theory of Robin Hood existing includes ballads that were found as far back as the 13th and 15th centuries. These ballads include stories of a commoner (not an outlaw) that lived in Sherwood Forest and fought with the Sheriff of Nottingham.
An Excerpt from an Epitaph by Thomas Gale (with translation)
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Reasons for a Lack of Evidence
Many of the reasons that there isn't much evidence to prove Robin Hood's existence is because the tales of Robin Hood were originally passed down orally (by voice). Each time that the tales are told and then repeated they are slightly modified and therefore no tale is told exactly the same twice.
Some historians, including Sidney Lee (author of the Dictionary of National Biography), have tried to write Robin Hood out of history. This is because they believe that Robin never existed.
There is also the chance that the legends of Robin Hood have obscured the truth so much that we are now unable to find the truth that may be within the tales of Robin Hood.