This is an age old bug, but one which to date has only been fixed by modders for PC. No console fix to date.

Linwe's Armor is a unique hooded leather armour set, acquirable from the NPC Linwe in the Thieves Guild Reputation Quest 'Summerset Shadows': http://www.uesp.net/wiki/Skyrim:Summerset_Shadows

It's a pretty awesome unique armour with a serious aesthetic flaw. In first person view, only when wearing this specific armour, your character's arms become invisible:

http://static.skyrim.nexusmods.com/mods/images/1956-2-1322531809.jpg

Like I say, there's a mod fix for PC, but on Xbox and PS3 you are kinda stuck with these weird floating arms.

As a secondary issue the enchantments applied to Linwe's Hood, Armor, Boots and Gloves are all enchanted, and can be disenchanted via an arcane enchanter. But it's a bit pointless, because the enchantments 'learned' cannot be used to enchant anything else in the game.