Introduction and Definition
Given two normed vector spaces V and W (over the same base field, either the real numbers R or the complex numbers C), a linear map A : V → W is continuous if and only if there exists a real number c such that
(the norm on the left is the one in W, the norm on the right is the one in V). Intuitively, the continuous operator A never "lengthens" any vector more than by a factor of c. Thus the image of a bounded set under a continuous operator is also bounded. Because of this property, the continuous linear operators are also known as bounded operators. In order to "measure the size" of A, it then seems natural to take the smallest number c such that the above inequality holds for all v in V. In other words, we measure the "size" of A by how much it "lengthens" vectors in the "biggest" case. So we define the operator norm of A as
(the minimum exists as the set of all such c is closed, nonempty, and bounded from below).
Read more about this topic: Operator Norm
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