Bounded operator

In <a href="/facts/Functional_analysis/MIp233MT">functional analysis</a> and <a href="/facts/Operator_theory/Ays1ofel">operator theory</a>, a bounded linear operator is a <a href="/facts/Linear_transformation/Q1PBKNVV">linear transformation</a> 
 
 
 
 L
 :
 X
 →
 Y
 
 
 {\displaystyle L:X\to Y}
 
 between <a href="/facts/Topological_vector_space/XRex1ChN">topological vector spaces</a> (TVSs) 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 that maps <a href="/facts/Bounded_set_(topological_vector_space)/bTsWeU9g">bounded</a> subsets of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 to bounded subsets of 
 
 
 
 Y
 .
 
 
 {\displaystyle Y.}
 
 
If 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 are <a href="/facts/Normed_vector_space/rmDljfyE">normed vector spaces</a> (a special type of TVS), then 
 
 
 
 L
 
 
 {\displaystyle L}
 
 is bounded if and only if there exists some 
 
 
 
 M
 >
 0
 
 
 {\displaystyle M>0}
 
 such that for all 
 
 
 
 x
 ∈
 X
 ,
 
 
 {\displaystyle x\in X,}

‖
        L
        x
        
          ‖
          
            Y
          
        
        ≤
        M
        ‖
        x
        
          ‖
          
            X
          
        
        .
      
    
    {\displaystyle \|Lx\|_{Y}\leq M\|x\|_{X}.}

The smallest such 
 
 
 
 M
 
 
 {\displaystyle M}
 
 is called the <a href="/facts/Operator_norm/a5l0TkFW">operator norm</a> of 
 
 
 
 L
 
 
 {\displaystyle L}
 
 and denoted by 
 
 
 
 ‖
 L
 ‖
 .
 
 
 {\displaystyle \|L\|.}
 
 
A linear operator between normed spaces is <a href="/facts/Continuous_linear_operator/KcqjDBrJ">continuous</a> if and only if it is bounded.
The concept of a bounded linear operator has been extended from normed spaces to all topological vector spaces.
Outside of functional analysis, when a function 
 
 
 
 f
 :
 X
 →
 Y
 
 
 {\displaystyle f:X\to Y}
 
 is called "<a href="/facts/Bounded_function/whVjbtuk">bounded</a>" then this usually means that its <a href="/facts/Image_of_a_function/KANefMAl">image</a> 
 
 
 
 f
 (
 X
 )
 
 
 {\displaystyle f(X)}
 
 is a bounded subset of its codomain. A linear map has this property if and only if it is identically 
 
 
 
 0.
 
 
 {\displaystyle 0.}
 
 
Consequently, in functional analysis, when a linear operator is called "bounded" then it is never meant in this abstract sense (of having a bounded image).

Bounded operator open-in-new

Bounded operator