Subspace of l2/L2 that is closed/not closed.

In summary, the conversation discusses the task of finding examples of infinite dimensional subspaces in l2(R) and L2(R), as well as determining whether these subspaces are closed or not. The examples mentioned include the set of sequences {1/n^p : n, p is N} for l2 and the normal distribution for L2, but it is unclear if these examples fulfill the criteria of being infinite dimensional and closed.
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Homework Statement



Give a nontrivial example of an infinite dimensional subspace in l2(R) that is closed. Also find an example of an infinite dimensional subspace of l2(R) that is not closed. Repeat the same two questions for L2(R).

The Attempt at a Solution



To my understanding, l2 is the space of square summable sequences and L2 is the space of square integrable functions. So basically we need to get finite sums / finite integrals to be in l2/L2.

For l2 I'm thinking of the set of sequences { 1/n^p : n, p is N }. I guess my problem is I don't really know if that is "infinite dimensional" and if it is closed or not.

Then for L2 I was thinking of the normal distribution because I know the area under it is finite. Again though, I'm not sure if it is infinite dimensional or closed.
 
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haha...posting yur 489 HW online...loser!
 

Related to Subspace of l2/L2 that is closed/not closed.

1. What is a subspace of l2/L2 that is closed/not closed?

A subspace of l2/L2 that is closed/not closed refers to a subset of the vector space l2/L2 that either includes all limit points or may not include all limit points, respectively. In other words, a closed subspace is one in which all convergent sequences of vectors in the subspace converge to a vector within the subspace, while a not closed subspace may have some convergent sequences that do not converge to a vector within the subspace.

2. How can I determine if a subspace of l2/L2 is closed/not closed?

To determine if a subspace of l2/L2 is closed or not closed, you can apply the definition of a closed subspace mentioned above. You can also use the criteria that a subspace is closed if and only if it contains all its boundary points, while a subspace is not closed if it has at least one boundary point not contained in the subspace.

3. What are some examples of closed subspaces of l2/L2?

Some examples of closed subspaces of l2/L2 include the space of all constant sequences, the space of all sequences that are eventually zero, and the space of all finite linear combinations of a given set of vectors. These examples satisfy the definition of a closed subspace, as all convergent sequences in these spaces converge to a vector within the space.

4. Can a subspace of l2/L2 be both closed and not closed?

No, a subspace of l2/L2 cannot be both closed and not closed. A subspace is either closed or not closed, based on the criteria mentioned above. However, it is possible for a subset of l2/L2 to be neither closed nor not closed, as it may not have any limit points at all.

5. How does the concept of a closed subspace relate to the concept of completeness?

A closed subspace is a subset of a complete space that is itself complete. In other words, a closed subspace is a complete space in its own right. This means that all Cauchy sequences within the subspace converge to a vector within the subspace, which is a key property of completeness. Therefore, the concept of a closed subspace is closely related to the concept of completeness in a vector space.

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