Is the Closure of a Subset in l^{1} Compact?

BobSun · 2008-12-05T02:32:47+00:00

Homework Statement Consider the Banach Space l^{1}. Let S={x \in l^{1}|\left\|x\right\|<1}. Is S a compact subset of l^{1}? prove or Disprove.

Thread starter BobSun
Start date Dec 4, 2008
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Banach Space

In summary, a Banach space is a complete normed vector space that is equipped with a norm function and is also complete in the sense that all Cauchy sequences converge to a limit within the space. Compactness is an important property of Banach spaces as it guarantees the existence of certain types of convergent sequences. It is defined as a subset that is both closed and bounded, meaning it contains all of its limit points and its elements have a finite distance from the origin. Examples of compact and non-compact subsets in Banach spaces include closed balls, finite sets, and subsets with finite dimensions, as well as open balls, infinite sets, and subsets with infinite dimensions. Compactness is also related to the Heine-Borel

Dec 4, 2008

BobSun

Homework Statement

Consider the Banach Space [tex]l^{1}[/tex]. Let S={[tex]x \in l^{1}|\left\|x\right\|<1[/tex]}. Is S a compact subset of [tex]l^{1}[/tex]? prove or Disprove.

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Dec 5, 2008

morphism

Science Advisor

Homework Helper

S isn't even closed. A more interesting problem is whether the closure of S is compact, and I suspect this is what you're supposed to work on.

1. What is a Banach space?

A Banach space is a complete normed vector space. This means that it is a vector space equipped with a norm function that assigns a non-negative length or size to each vector in the space, and it is also complete in the sense that all Cauchy sequences converge to a limit within the space.

2. What is the importance of compactness in Banach spaces?

Compactness is an important property of Banach spaces because it guarantees the existence of certain types of convergent sequences within the space. This property is crucial in many areas of mathematics, including functional analysis, topology, and differential equations.

3. How is compactness defined in Banach spaces?

A subset of a Banach space is considered compact if it is both closed and bounded. This means that the subset contains all of its limit points and all of its elements have a finite distance from the origin.

4. What are some examples of compact and non-compact subsets in Banach spaces?

Examples of compact subsets in Banach spaces include closed balls, finite sets, and subsets with finite dimensions. Non-compact subsets can include open balls, infinite sets, and subsets with infinite dimensions.

5. How is compactness related to the Heine-Borel theorem?

The Heine-Borel theorem states that a subset of Euclidean space is compact if and only if it is closed and bounded. This theorem can also be extended to Banach spaces, where a subset is compact if and only if it is closed and bounded in the norm topology.