Norms 2.4 2002/07/31 2006/08/02 14:34:00.015 GMT-5 Michael Haag mjhaag@rice.edu Justin Romberg jrom@rice.edu Justin Romberg jrom@rice.edu Richard G. Baraniuk richb@rice.edu Michael Haag mjhaag@rice.edu Mariyah Poonawala mariyah@rice.edu Prashant Singh prash@ece.rice.edu Matthew Hutchinson mhutch@rice.edu Euclidean norm norm normalization normed linear space normed vector space norms This module will define a norm and give examples and properties of it.

Introduction Much of the language in this section will be familiar to you - you should have previously been exposed to the concepts of inner products orthogonality basis expansions in the context of n . We're going to take what we know about vectors and apply it to functions (continuous time signals).

Norms The norm of a vector is a real number that represents the "size" of the vector. In 2 , we can define a norm to be a vectors geometric length.

x x 0 x 1 , norm x x 0 2 x 1 2 Mathematically, a norm · is just a function (taking a vector and returning a real number) that satisfies three rules. To be a norm, · must satisfy: the norm of every vector is positive x x S x 0 scaling a vector scales the norm by the same amount α x α x for all vectors x and scalars α Triangle Property: x y x y for all vectors x , y . "The "size" of the sum of two vectors is less than or equal to the sum of their sizes" A vector space with a well defined norm is called a normed vector space or normed linear space.

Examples n (or n ), x x 0 x 1 … x n - 1 , 1 x i 0 n 1 x i , n with this norm is called ℓ 1 ( [ 0 , n - 1 ] ) .

Collection of all x 2 with 1 x 1

n (or n ), with norm 2 x i 0 n 1 x i 2 1 2 , n is called ℓ 2 ( [ 0 , n - 1 ] ) (the usual "Euclidean"norm).

Collection of all x 2 with 2 x 1

n (or n , with norm x i x i is called ℓ ∞ ( [ 0 , n - 1 ] )

x 2 with x 1

Spaces of Sequences and Functions We can define similar norms for spaces of sequences and functions. Discrete time signals = sequences of numbers x n … x -2 x -1 x 0 x 1 x 2 … 1 x n i x i , x n ℓ 1 ( ℤ ) 1 x 2 x n i x i 2 1 2 , x n ℓ 2 ( ℤ ) 2 x p x n i x i p 1 p , x n ℓ p ( ℤ ) p x x n sup i | x [ i ] | , x n ℓ ∞ ( ℤ ) x For continuous time functions: p f t t f t p 1 p , f t L p ( ℝ ) p f t (On the interval) p f t t 0 T f t p 1 p , f t L p ( [ 0 , T ] ) p f t