Construct a continuous linear map from a linear map and a bound on this linear map.

A continuous linear map between normed spaces is bounded when the field is nondiscrete. The continuity ensures boundedness on a ball of some radius δ. The nondiscreteness is then used to rescale any element into an element of norm in [δ/C, δ], whose image has a controlled norm. The norm control for the original element follows by rescaling.

The operator norm of a continuous linear map is the inf of all its bounds.

The fundamental property of the operator norm: ∥f x∥ ≤ ∥f∥ * ∥x∥.

The image of the unit ball under a continuous linear map is bounded.

If one controls the norm of every A x, then one controls the norm of A.

The operator norm satisfies the triangle inequality.

An operator is zero iff its norm vanishes.

The norm of the identity is at most 1. It is in fact 1, except when the space is trivial where it is 0. It means that one can not do better than an inequality in general.

The operator norm is homogeneous.

Continuous linear maps themselves form a normed space with respect to the operator norm.

The operator norm is submultiplicative.

continuous linear maps are Lipschitz continuous.

The norm of the tensor product of a scalar linear map and of an element of a normed space is the product of the norms.