In
mathematics, one
normed vector space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length ...
is said to be continuously embedded in another normed vector space if the
inclusion function
In mathematics, if A is a subset of B, then the inclusion map (also inclusion function, insertion, or canonical injection) is the function \iota that sends each element x of A to x, treated as an element of B:
\iota : A\rightarrow B, \qquad \iota ...
between them is
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
. In some sense, the two norms are "almost equivalent", even though they are not both defined on the same space. Several of the
Sobolev embedding theorems are continuous embedding theorems.
Definition
Let ''X'' and ''Y'' be two normed vector spaces, with norms , , ·, ,
''X'' and , , ·, ,
''Y'' respectively, such that ''X'' ⊆ ''Y''. If the
inclusion map (identity function)
:
is continuous, i.e. if there exists a constant ''C'' > 0 such that
:
for every ''x'' in ''X'', then ''X'' is said to be continuously embedded in ''Y''. Some authors use the hooked arrow "↪" to denote a continuous embedding, i.e. "''X'' ↪ ''Y''" means "''X'' and ''Y'' are normed spaces with ''X'' continuously embedded in ''Y''". This is a consistent use of notation from the point of view of the
category of topological vector spaces, in which the
morphisms ("arrows") are the
continuous linear map In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces.
An operator between two normed spaces is a bounded linear o ...
s.
Examples
* A finite-dimensional example of a continuous embedding is given by a natural embedding of the
real line ''X'' = R into the plane ''Y'' = R
2, where both spaces are given the Euclidean norm:
::
:In this case, , , ''x'', ,
''X'' = , , ''x'', ,
''Y'' for every real number ''X''. Clearly, the optimal choice of constant ''C'' is ''C'' = 1.
* An infinite-dimensional example of a continuous embedding is given by the
Rellich–Kondrachov theorem In mathematics, the Rellich–Kondrachov theorem is a compact embedding theorem concerning Sobolev spaces. It is named after the Austrian-German mathematician Franz Rellich and the Russian mathematician Vladimir Iosifovich Kondrashov. Rellich pr ...
: let Ω ⊆ R
''n'' be an
open
Open or OPEN may refer to:
Music
* Open (band), Australian pop/rock band
* The Open (band), English indie rock band
* ''Open'' (Blues Image album), 1969
* ''Open'' (Gotthard album), 1999
* ''Open'' (Cowboy Junkies album), 2001
* ''Open'' ( ...
,
bounded,
Lipschitz domain In mathematics, a Lipschitz domain (or domain with Lipschitz boundary) is a domain in Euclidean space whose boundary is "sufficiently regular" in the sense that it can be thought of as locally being the graph of a Lipschitz continuous function. T ...
, and let 1 ≤ ''p'' < ''n''. Set
::
:Then the Sobolev space ''W''
1,''p''(Ω; R) is continuously embedded in the
''L''''p'' space ''L''
''p''∗(Ω; R). In fact, for 1 ≤ ''q'' < ''p''
∗, this embedding is
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
. The optimal constant ''C'' will depend upon the geometry of the domain Ω.
* Infinite-dimensional spaces also offer examples of ''discontinuous'' embeddings. For example, consider
::
:the space of continuous real-valued functions defined on the unit interval, but equip ''X'' with the ''L''
1 norm and ''Y'' with the
supremum norm
In mathematical analysis, the uniform norm (or ) assigns to real- or complex-valued bounded functions defined on a set the non-negative number
:\, f\, _\infty = \, f\, _ = \sup\left\.
This norm is also called the , the , the , or, when th ...
. For ''n'' ∈ N, let ''f''
''n'' be the
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
,
piecewise linear function given by
::
:Then, for every ''n'', , , ''f''
''n'', ,
''Y'' = , , ''f''
''n'', ,
∞ = ''n'', but
::
:Hence, no constant ''C'' can be found such that , , ''f''
''n'', ,
''Y'' ≤ ''C'', , ''f''
''n'', ,
''X'', and so the embedding of ''X'' into ''Y'' is discontinuous.
See also
*
Compact embedding In mathematics, the notion of being compactly embedded expresses the idea that one set or space is "well contained" inside another. There are versions of this concept appropriate to general topology and functional analysis.
Definition (topological ...
References
* {{cite book , author1=Rennardy, M. , author2= Rogers, R.C. , name-list-style=amp , title=An Introduction to Partial Differential Equations , publisher=Springer-Verlag, Berlin , year=1992 , isbn=3-540-97952-2
Functional analysis