##  [Sobolev Space](/sobolev-space) 

  ##  [Sobolev Space](https://natural.quantumdictionary.io/sobolev-space-0) 

  

 [![Natural & Formal Sciences Dictionary](/sites/default/files/styles/large/public/2026-01/Natural%20%26%20Formal%20Sciences.png.webp?itok=2kCDRVQv)](/topic-specific-dictionaries/natural-formal-sciences)



**Natural & Formal Sciences Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A Banach (or Hilbert) space of equivalence classes of functions on a domain whose weak derivatives up to a specified order belong to an Lp space; commonly denoted W^{k,p} or H^{k} for p=2.

 

 

 

 

 





 

 



 ##  [Sobolev Space](https://mathlogic.quantumdictionary.io/sobolev-space-1) 

  

 [![Mathematics & Logic Dictionary](/sites/default/files/styles/large/public/2026-01/Mathematics%20%26%20Logic.png.webp?itok=UhtTRPnp)](/topic-specific-dictionaries/natural-formal-sciences/mathematics-logic)

- Natural & Formal Sciences -

**Mathematics & Logic Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A family of function spaces W^{k,p} (or H^s when p=2) that measure both p‑integrability and weak differentiability up to integer order k (or fractional order s) by equipping functions with norms combining L^p norms of the function and its weak derivatives.

 

 

 

 

 





 

 



 ##  [Sobolev Space](https://puremath.quantumdictionary.io/sobolev-space-2) 

  

 [![Pure Mathematics Dictionary](/sites/default/files/styles/large/public/2026-01/Pure%20Mathematics.png.webp?itok=5pZnFQ59)](/topic-specific-dictionaries/mathematics-logic/pure-mathematics)

- Mathematics & Logic -

**Pure Mathematics Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A Banach or Hilbert space of functions or distributions on a domain endowed with a norm that measures integrability and the size of weak derivatives up to a specified order; typically denoted W^{k,p} or H^s and fundamental to variational formulations and PDE theory.