operator norm (original) (raw)

Definition

Let A:𝖵→𝖶 be a linear map between normed vector spaces 𝖵 and𝖶. To each such map (operator) A we can assign a non-negative number∥A∥op defined by

where the supremum ∥A∥op could be finite or infinite. Equivalently, the above definition can be written as

By convention, if 𝖵 is the zero vector space, any operator from 𝖵to 𝖶 must be the zero operator and is assigned zero norm.

∥A∥op is called the the operator norm (or the induced norm) of A, for reasons that will be clear in the next .

Operator norm is in fact a norm

Definition - If ∥A∥op is finite, we say that A is a. Otherwise, we say that A is .

It turns out that, for bounded operators, ∥⋅∥op satisfies all the properties of a norm (hence the name operator norm). The proof follows immediately from the definition:

Positivity:

Since ∥A⁢𝐯∥≥0, by definition∥A∥op≥0. Also, ∥A⁢𝐯∥=0 identically only if A=0. Hence ∥A∥op=0 only if A=0.

Absolute homogeneity:

Since ∥λ⁢A⁢𝐯∥=|λ|⁢∥A⁢𝐯∥, by definition∥λ⁢A∥op=|λ|⁢∥A∥op.

Triangle inequality:

Since ∥(A+B)⁢𝐯∥=∥A⁢𝐯+B⁢𝐯∥≤∥A⁢𝐯∥+∥B⁢𝐯∥, by definition∥A+B∥op≤∥A∥op+∥B∥op.

The set L⁢(𝖵,𝖶) of bounded linear maps from 𝖵 to 𝖶 forms a vector space and ∥⋅∥op defines a norm in it.

Example

Suppose that 𝖵=(ℝn,∥⋅∥p) and 𝖶=(ℝn,∥⋅∥p), where∥⋅∥p is the vector p-norm. Then the operator norm∥⋅∥op=∥⋅∥p is the matrix p-norm.