## Advances in Operator Theory

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

### On orthogonal decomposition of a Sobolev space

#### Abstract

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

**Source**

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

**Dates**

Received: 11 March 2017

Accepted: 8 June 2017

First available in Project Euclid: 4 December 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.aot/1512431718

**Digital Object Identifier**

doi:10.22034/aot.1703-1135

**Mathematical Reviews number (MathSciNet)**

MR3730037

**Zentralblatt MATH identifier**

1385.46023

**Subjects**

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

Secondary: 46C15: Characterizations of Hilbert spaces

**Keywords**

Sobolev space orthogonal decomposition inner product distance

#### Citation

Lakew, Dejenie. On orthogonal decomposition of a Sobolev space. Adv. Oper. Theory 2 (2017), no. 4, 419--427. doi:10.22034/aot.1703-1135. https://projecteuclid.org/euclid.aot/1512431718