Those jihadists , they're all about culture and learning aren't they. Hey, maybe 0bama can use NASA to teach everyone all about the contributions they made to space exploration. </s>
Yes, but there is considerable debate about how much was left to destroy by that point.
Numerous wars, insurrections, earthquakes, etc. over the centuries, and transfer of contents to Rome and Constantinople.
Also the early Christians of Alexandria were particularly feisty and reportedly did at least some damage to the pagan Library and Museum.
http://en.wikipedia.org/wiki/Library_of_Alexandria#Different_Theories_Regarding_Destruction_of_the_Library
Part of the reason it is contentious is that various groups want to blame other groups they dislike for the destruction of the Library.