Bounded operator

In functional analysis, a bounded linear operator is a linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L(v) to that of v is bounded above by the same number, over all non-zero vectors v in X. In other words, there exists some M ≥ 0 {displaystyle Mgeq 0} such that for all v in X In functional analysis, a bounded linear operator is a linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L(v) to that of v is bounded above by the same number, over all non-zero vectors v in X. In other words, there exists some M ≥ 0 {displaystyle Mgeq 0} such that for all v in X The smallest such M is called the operator norm ‖ L ‖ o p {displaystyle |L|_{mathrm {op} },} of L. A bounded linear operator is generally not a bounded function, as generally one can find a sequence v k {displaystyle v_{k}} in X such that ‖ L v k ‖ Y → ∞ {displaystyle |Lv_{k}|_{Y} ightarrow infty } . Instead, all that is required for the operator to be bounded is that So, the operator L could only be a bounded function if it satisfied L(v)=0 for all v, as is easy to understand by considering that for a linear operator, L ( a v ) = a L ( v ) {displaystyle L(av)=aL(v)} for all scalars a.Rather, a bounded linear operator is a locally bounded function. A linear operator between normed spaces is bounded if and only if it is continuous, and by linearity, if and only if it is continuous at zero. As stated in the introduction, a linear operator L between normed spaces X and Y is bounded if and only if it is a continuous linear operator. The proof is as follows. Not every linear operator between normed spaces is bounded. Let X be the space of all trigonometric polynomials defined on , with the norm Define the operator L:X→X which acts by taking the derivative, so it maps a polynomial P to its derivative P′. Then, for with n=1, 2, ...., we have ‖ v ‖ = 2 π , {displaystyle |v|=2pi ,} while ‖ L ( v ) ‖ = 2 π n → ∞ {displaystyle |L(v)|=2pi n o infty } as n→∞, so this operator is not bounded.