Balanced set

In <a href="/facts/Linear_algebra/8VaB4VWk">linear algebra</a> and related areas of <a href="/facts/Mathematics/pxTouaz4">mathematics</a> a balanced set, circled set or disk in a <a href="/facts/Vector_space/M1pxMLx2">vector space</a> (over a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> 
 
 
 
 
 K
 
 
 
 {\displaystyle \mathbb {K} }
 
 with an <a href="/facts/Absolute_value_(algebra)/UUncm7y0">absolute value</a> function 
 
 
 
 
 |
 
 ⋅
 
 |
 
 
 
 {\displaystyle |\cdot |}
 
) is a <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> 
 
 
 
 S
 
 
 {\displaystyle S}
 
 such that 
 
 
 
 a
 S
 ⊆
 S
 
 
 {\displaystyle aS\subseteq S}
 
 for all <a href="/facts/Scalar_(mathematics)/MkDbA4Eo">scalars</a> 
 
 
 
 a
 
 
 {\displaystyle a}
 
 satisfying 
 
 
 
 
 |
 
 a
 
 |
 
 ≤
 1.
 
 
 {\displaystyle |a|\leq 1.}

The balanced hull or balanced envelope of a set 
 
 
 
 S
 
 
 {\displaystyle S}
 
 is the smallest balanced set containing 
 
 
 
 S
 .
 
 
 {\displaystyle S.}

The balanced core of a set 
  
    
      
        S
      
    
    {\displaystyle S}
  
 is the largest balanced set contained in 
  
    
      
        S
        .
      
    
    {\displaystyle S.}

Balanced sets are ubiquitous in <a href="/facts/Functional_analysis/MIp233MT">functional analysis</a> because every <a href="/facts/Neighbourhood_(mathematics)/XHKgrpkp">neighborhood</a> of the origin in every <a href="/facts/Topological_vector_space/XRex1ChN">topological vector space</a> (TVS) contains a balanced neighborhood of the origin and every convex neighborhood of the origin contains a balanced convex neighborhood of the origin (even if the TVS is not <a href="/facts/Locally_convex_topological_vector_space/AGrPrayC">locally convex</a>). This neighborhood can also be chosen to be an <a href="/facts/Open_set/QfTCIqMu">open set</a> or, alternatively, a <a href="/facts/Closed_set/upYnRTUj">closed set</a>.

Balanced set open-in-new

Balanced set