In mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its operator norm. Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces. Informally, the operator norm ‖ T ‖ {\displaystyle \|T\|} of a linear map T : X → Y {\displaystyle T:X\to Y} is the maximum factor by which it "lengthens" vectors. It is also called the bound norm.
Introduction and definition Given two normed vector spaces V {\displaystyle V} and W {\displaystyle W} (over the same base field, either the real numbers R {\displaystyle \mathbb {R} } or the complex numbers C {\displaystyle \mathbb {C} } ), a linear map A : V → W {\displaystyle A:V\to W} is continuous if and only if there exists a real number c {\displaystyle c} such that
‖ A v ‖ ≤ c ‖ v ‖ for all v ∈ V . {\displaystyle \|Av\|\leq c\|v\|\quad {\text{ for all }}v\in V.}
The norm on the left is the one in W {\displaystyle W} and the norm on the right is the one in V {\displaystyle V} . Intuitively, the continuous operator A {\displaystyle A} never increases the length of any vector by more than a factor of c . {\displaystyle 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 , {\displaystyle A,} one can take the infimum of the numbers c {\displaystyle c} such that the above inequality holds for all v ∈ V . {\displaystyle v\in V.} This number represents the maximum scalar factor by which A {\displaystyle A} "lengthens" vectors. In other words, the "size" of A {\displaystyle A} is measured by how much it "lengthens" vectors in the "biggest" case. So we define the operator norm of A {\displaystyle A} as
‖ A ‖ op = inf { c ≥ 0 : ‖ A v ‖ ≤ c ‖ v ‖ for all v ∈ V } . {\displaystyle \|A\|_{\text{op}}=\inf\{c\geq 0:\|Av\|\leq c\|v\|{\text{ for all }}v\in V\}.}
The infimum is attained as the set of all such c {\displaystyle 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 {\displaystyle V} and W {\displaystyle W} .
Examples Every real m {\displaystyle m} -by- n {\displaystyle n} matrix corresponds to a linear map from R n {\displaystyle \mathbb {R} ^{n}} to R m . {\displaystyle \mathbb {R} ^{m}.} Each pair of the plethora of (vector) norms applicable to real vector spaces induces an operator norm for all m {\displaystyle m} -by- n {\displaystyle n} matrices of real numbers; these induced norms form a subset of matrix norms. If we specifically choose the Euclidean norm on both R n {\displaystyle \mathbb {R} ^{n}} and R m , {\displaystyle \mathbb {R} ^{m},} then the matrix norm given to a matrix A {\displaystyle A} is the square root of the largest eigenvalue of the matrix A ∗ A {\displaystyle A^{*}A} (where A ∗ {\displaystyle A^{*}} denotes the conjugate transpose of A {\displaystyle A} ). This is equivalent to assigning the largest singular value of A . {\displaystyle A.}
Passing to a typical infinite-dimensional example, consider the sequence space ℓ 2 , {\displaystyle \ell ^{2},} which is an Lp space, defined by
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