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A set-theoretic inclusion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050200/i0502001.png" /> of a linear [[Normed space|normed space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050200/i0502002.png" /> into a linear normed space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050200/i0502003.png" />, for which the following inequality is valid for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050200/i0502004.png" />:
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050200/i0502005.png" /></td> </tr></table>
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050200/i0502006.png" /> is a constant which does not depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050200/i0502007.png" />. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050200/i0502008.png" /> is the norm (semi-norm) of the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050200/i0502009.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050200/i05020010.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050200/i05020011.png" /> is the norm (semi-norm) of the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050200/i05020012.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050200/i05020013.png" />.
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A set-theoretic inclusion  $  V \subset  W $
 +
of a linear [[Normed space|normed space]]  $  V $
 +
into a linear normed space  $  W $,
 +
for which the following inequality is valid for any  $  x \in V $:
  
The identity operator from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050200/i05020014.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050200/i05020015.png" />, which assigns to an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050200/i05020016.png" /> the same element seen as an element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050200/i05020017.png" />, is said to be the imbedding operator of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050200/i05020018.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050200/i05020019.png" />. The imbedding operator is always bounded. If the imbedding operator is a [[Completely-continuous operator|completely-continuous operator]], the imbedding of function spaces is said to be compact. Facts on imbedding of function spaces are established by so-called [[Imbedding theorems|imbedding theorems]].
+
$$
 +
\| x \| _ {W}  \leq  C  \| x \| _ {V} ,
 +
$$
  
Example. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050200/i05020020.png" /> be a Lebesgue-measurable set in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050200/i05020021.png" />-dimensional Euclidean space with finite measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050200/i05020022.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050200/i05020023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050200/i05020024.png" />, be the Lebesgue space of measurable functions which are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050200/i05020025.png" />-th power summable over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050200/i05020026.png" /> with norm
+
where  $  C $
 +
is a constant which does not depend on  $  x \in V $.
 +
Here,  $  \| x \| _ {W} $
 +
is the norm (semi-norm) of the element  $  x $
 +
in  $  W $,  
 +
while  $  \| x \| _ {V} $
 +
is the norm (semi-norm) of the element  $  x $
 +
in  $  V $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050200/i05020027.png" /></td> </tr></table>
+
The identity operator from  $  V $
 +
into  $  W $,
 +
which assigns to an element  $  x \in V $
 +
the same element seen as an element of  $  W $,
 +
is said to be the imbedding operator of  $  V $
 +
into  $  W $.
 +
The imbedding operator is always bounded. If the imbedding operator is a [[Completely-continuous operator|completely-continuous operator]], the imbedding of function spaces is said to be compact. Facts on imbedding of function spaces are established by so-called [[Imbedding theorems|imbedding theorems]].
  
Then, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050200/i05020028.png" />, one has the imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050200/i05020029.png" />, and
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Example. Let  $  E $
 +
be a Lebesgue-measurable set in the  $  n $-
 +
dimensional Euclidean space with finite measure  $  \mathop{\rm mes}  E $
 +
and let  $  L _ {p} ( E) $,  
 +
$  1 \leq  p \leq  \infty $,  
 +
be the Lebesgue space of measurable functions which are  $  p $-
 +
th power summable over  $  E $
 +
with norm
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050200/i05020030.png" /></td> </tr></table>
+
$$
 +
\| x \| _ {p}  = \
 +
\left [ \int\limits _ { E }
 +
| x ( t) |  ^ {p}
 +
dt \right ]  ^ {1/p} .
 +
$$
  
 +
Then, if  $  p \geq  q $,
 +
one has the imbedding  $  L _ {p} ( E) \rightarrow L _ {q} ( E) $,
 +
and
  
 +
$$
 +
\| x \| _ {q}  \leq  \
 +
(  \mathop{\rm mes}  E) ^ {1/q- 1/p }
 +
\| x \| _ {p} .
 +
$$
  
 
====Comments====
 
====Comments====
 
For references cf. [[Imbedding theorems|Imbedding theorems]].
 
For references cf. [[Imbedding theorems|Imbedding theorems]].

Latest revision as of 22:11, 5 June 2020


A set-theoretic inclusion $ V \subset W $ of a linear normed space $ V $ into a linear normed space $ W $, for which the following inequality is valid for any $ x \in V $:

$$ \| x \| _ {W} \leq C \| x \| _ {V} , $$

where $ C $ is a constant which does not depend on $ x \in V $. Here, $ \| x \| _ {W} $ is the norm (semi-norm) of the element $ x $ in $ W $, while $ \| x \| _ {V} $ is the norm (semi-norm) of the element $ x $ in $ V $.

The identity operator from $ V $ into $ W $, which assigns to an element $ x \in V $ the same element seen as an element of $ W $, is said to be the imbedding operator of $ V $ into $ W $. The imbedding operator is always bounded. If the imbedding operator is a completely-continuous operator, the imbedding of function spaces is said to be compact. Facts on imbedding of function spaces are established by so-called imbedding theorems.

Example. Let $ E $ be a Lebesgue-measurable set in the $ n $- dimensional Euclidean space with finite measure $ \mathop{\rm mes} E $ and let $ L _ {p} ( E) $, $ 1 \leq p \leq \infty $, be the Lebesgue space of measurable functions which are $ p $- th power summable over $ E $ with norm

$$ \| x \| _ {p} = \ \left [ \int\limits _ { E } | x ( t) | ^ {p} dt \right ] ^ {1/p} . $$

Then, if $ p \geq q $, one has the imbedding $ L _ {p} ( E) \rightarrow L _ {q} ( E) $, and

$$ \| x \| _ {q} \leq \ ( \mathop{\rm mes} E) ^ {1/q- 1/p } \| x \| _ {p} . $$

Comments

For references cf. Imbedding theorems.

How to Cite This Entry:
Imbedding of function spaces. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Imbedding_of_function_spaces&oldid=11516
This article was adapted from an original article by L.P. Kuptsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article