Operator norm

In mathematics, the operator norm is a means to measure the 'size' of certain linear operators. Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces. In mathematics, the operator norm is a means to measure the 'size' of certain linear operators. Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces. 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 and the norm on the right is the one in V. Intuitively, the continuous operator A never increases the length of 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 infimum of the numbers 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 infimum is attained as the set of all such c is closed, nonempty, and bounded from below. It is important to bear in mind that this operator norm depends on the choice of norms for the normed vector spaces V and W. Every real m-by-n matrix corresponds to a linear map from Rn to Rm. Each pair of the plethora of (vector) norms applicable to real vector spaces induces an operator norm for all m-by-n matrices of real numbers; these induced norms form a subset of matrix norms. If we specifically choose the Euclidean norm on both Rn and Rm, then we obtain the matrix norm which to a given matrix A assigns the square root of the largest eigenvalue of the matrix A*A (where A* denotes the conjugate transpose of A). This is equivalent to assigning the largest singular value of A. Passing to a typical infinite-dimensional example, consider the sequence space l 2 {displaystyle l^{2}} defined by This can be viewed as an infinite-dimensional analogue of the Euclidean space Cn. Now take a bounded sequences = (sn ). The sequence s is an element of the space l ∞, with a norm given by