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Normed Vector Space

Module by: Phil Schniter. E-mail the author

Summary: This module introduces normed vector space.

Now we equip a vector space VV with a notion of "size".

  • A norm is a function ( ∥⋅∥:V→R : ⋅ V ) such that the following properties hold (   ,   x−∈V&y−∈V    x y x V y V and   ,   α∈    α α ):
    1. ∥x−∥≥0 x 0 with equality iff x−=0 x 0
    2. ∥α⁢x−∥=|α|⁢⋅⁢∥x−∥ α x α ⋅ x
    3. ∥x−+y−∥≤∥x−∥+∥y−∥ x y x y , (the triangle inequality).
    In simple terms, the norm measures the size of a vector. Adding the norm operation to a vector space yields a normed vector space. Important example include:
    1. V=RN V N , ∥ x 0 … x N - 1 T∥≔∑ i =0N−1 x i 2≔x−T⁢x− ≔ x 0 … x N - 1 i 0 N 1 x i 2 x x
    2. V=CN V N , ∥ x 0 … x N - 1 T∥≔∑ i =0N−1| x i |2≔x−H⁢x− ≔ x 0 … x N - 1 i 0 N 1 x i 2 x x
    3. V= l p V l p , ∥x⁢n∥≔∑n=−∞∞|x⁢n|p1p ≔ x n n x n p 1 p
    4. V= ℒ p V ℒ p , ∥f⁢t∥≔∫−∞∞|f⁢t|pdt 1p ≔ f t t f t p 1 p

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