bounded operator

Definition [1]

  1. 1.

    Suppose X and Y are normed vector spacesPlanetmathPlanetmath with norms X and Y. Further, suppose T is a linear map T:XY. If there is a C𝐑 such that for all xX we have

    TxY CxX,

    then T is a bounded operatorMathworldPlanetmathPlanetmath.

  2. 2.

    Let X and Y be as above, and let T:XY be a bounded operator. Then the norm of T is defined as the real number


    Thus the operator norm is the smallest constant C𝐑 such that

    TxY CxX.

    Now for any xX{0}, if we let y=x/x, then linearity implies that


    and thus it easily follows that

    T = sup{TxYxX|xX{0}}=sup{TyY|xX{0},y=xx}
    = sup{TyY|yX,y=1}.

    In the special case when X={𝟎} is the zero vector space, any linear map T:XY is the zero map since T(𝟎)=T(𝟎𝟎)=0T(𝟎)=0. In this case, we define T:=0.

  3. 3.

    To avoid cumbersome notational stuff usually one can simplify the symbols like ||x||X and ||Tx||Y by writing only ||x||, ||Tx|| since there is a little danger in confusing which is space about calculating norms.

0.0.1 TO DO:

  1. 1.

    The defined norm for mappings is a norm

  2. 2.

    Examples: identity operator, zero operator: see [1].

  3. 3.

    Give alternative expressions for norm of T.

  4. 4.

    Discuss boundedness and continuity

TheoremMathworldPlanetmath [1, 2] Suppose T:XY is a linear map between normed vector spaces X and Y. If X is finite-dimensional, then T is boundedPlanetmathPlanetmathPlanetmathPlanetmath.

Theorem Suppose T:XY is a linear map between normed vector spaces X and Y. The following are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath:

  1. 1.

    T is continuousMathworldPlanetmathPlanetmath in some point x0X

  2. 2.
  3. 3.

    T is bounded

Lemma Any bounded operator with a finite dimensional kernel and cokernel has a closed image.

Proof By Banach’s isomorphism theorem.


  • 1 E. Kreyszig, Introductory Functional AnalysisMathworldPlanetmathPlanetmath With Applications, John Wiley & Sons, 1978.
  • 2 G. Bachman, L. Narici, Functional analysis, Academic Press, 1966.
Title bounded operator
Canonical name BoundedOperator
Date of creation 2013-03-22 14:02:17
Last modified on 2013-03-22 14:02:17
Owner bwebste (988)
Last modified by bwebste (988)
Numerical id 19
Author bwebste (988)
Entry type Definition
Classification msc 46B99
Related topic VectorNorm