Open mapping theorem (functional analysis)

<p>In <a href="/facts/Functional_analysis/MIp233MT">functional analysis</a>, the open mapping theorem, also known as the Banach–Schauder theorem or the Banach theorem (named after <a href="/facts/Stefan_Banach/b1qHSNg2">Stefan Banach</a> and <a href="/facts/Juliusz_Schauder/VSpCQfY1">Juliusz Schauder</a>), is a fundamental result that states that if a <a href="/facts/Bounded_linear_operator/LYmDTIqR">bounded</a> or <a href="/facts/Continuous_linear_operator/KcqjDBrJ">continuous linear operator</a> between <a href="/facts/Banach_space/sDEIk7EC">Banach spaces</a> is <a href="/facts/Surjective/KezhLpCb">surjective</a> then it is an <a href="/facts/Open_map/abHUKX7l">open map</a>.
</p><p>A special case is also called the bounded inverse theorem (also called inverse mapping theorem or Banach isomorphism theorem), which states that a bijective bounded linear operator 
  
    
      
        T
      
    
    {\displaystyle T}
  
 from one Banach space to another has bounded inverse 
  
    
      
        
          T
          
            −
            1
          
        
      
    
    {\displaystyle T^{-1}}
  
.
</p>

Open mapping theorem (functional analysis) open-in-new

Open mapping theorem (functional analysis)