Advances in Operator Theory

On orthogonal decomposition of a Sobolev space

Dejenie Lakew

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The theme of this short article is to investigate an orthogonal decomposition of the Sobolev space $W^{1,2}\left( \Omega \right) $ as $$ W^{1,2}\left( \Omega \right) =A^{2,2}\left( \Omega \right) \oplus D^{2}\left( W_{0}^{3,2}\left( \Omega \right) \right)$$ and look at some of the properties of the inner product therein and the distance defined from the inner product. We also determine the dimension of the orthogonal difference space $W^{1,2}\left( \Omega \right) \ominus \left(W_{0}^{1,2}\left( \Omega \right) \right) ^{\perp }$ and show the expansion of Sobolev spaces as their regularity increases.

Article information

Adv. Oper. Theory, Volume 2, Number 4 (2017), 419-427.

Received: 11 March 2017
Accepted: 8 June 2017
First available in Project Euclid: 4 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems
Secondary: 46C15: Characterizations of Hilbert spaces

Sobolev space orthogonal decomposition inner product distance


Lakew, Dejenie. On orthogonal decomposition of a Sobolev space. Adv. Oper. Theory 2 (2017), no. 4, 419--427. doi:10.22034/aot.1703-1135.

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  • D. A. Lakew, New proofs on properties of an orthogonal decomposition of a Hilbert space, $arXiv:1510.07944v1$.
  • D. A. Lakew, On Orthogonal decomposition of $L^2(\Omega)$, J. Math. Comput. Sci. 10 (2015), no. 1, 27–37.