White Sword

In The Legend of Zelda, the White Sword is the second sword which Link receives, following the Wooden Sword and before the Magical Sword. It can be obtained once Link has five hearts in his heart meter. It is obtained by speaking to the Old Man in a cave in the North-East of Hyrule, guarded by a Lynel. The Old Man will ask "Have you looked behind the waterfall?", which is a clue to the hiding place of the White Sword.

Elemental White Sword
In The Minish Cap, the White Sword is the reforged Picori Blade. It was infused with the four Elements, and grew in power as it was infused with the power of more Elements. Three elements made it the empowered White Sword, and all four elements made it into the Four Sword.

In the Japanese versions of Oracle of Seasons and Oracle of Ages, the White Sword is the original name of the Noble Sword.

Despite these two swords sharing the same name, the nature of the White Sword in The Minish Cap makes it unlikely that it is the same White Sword in The Legend of Zelda.