Examples
Every real m-by-n matrix yields a linear map from Rn to Rm. One can put several different norms on these spaces, as explained in the article on norms. Each such choice of norms gives rise to an operator norm and therefore yields a norm on the space of all m-by-n matrices. Examples can be found in the article on 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 defined by
This can be viewed as an infinite dimensional analogue of the Euclidean space Cn. Now take a bounded sequence s = (sn ). The sequence s is an element of the space l ∞, with a norm given by
Define an operator Ts by simply multiplication:
The operator T s is bounded with operator norm
One can extend this discussion directly to the case where l 2 is replaced by a general Lp space with p > 1 and l∞ replaced by L∞.
Read more about this topic: Operator Norm
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