As you know, Link's first weapon is nothing more than a stick, and he begins the game with no equipment. Surely Zelda, who has at least a few armed guards at her disposal, would be able to provide the Hero with something more substantial?
So, here's how it goes: when Link meets Zelda for the first time, he has access to various treasure chests in the East Palace, which contain:
* Magic Ring L1
* Worn-out Glove
* The first Bomb Bag
He also gets the L2 sword after talking to Zelda.
However, after leaving the East Palace... a gang of Lynels confront Link, and in cutscene, demand that he hand over the sword and all of his rupees. Link, being at this point a little wet-behind-the-ears, obliges. It's only after they've left that he realises that he could've just killed them with the sword. "Darn!"
(He can get the L2 sword back by invading the Lynel's lair, which is a decrepit Hyrule Castle.)
