Showing S1+S2 is Dense in Hilbert Space

In summary, the conversation discusses the orthonormal basis {ej} of a Hilbert space, and how to show that the sum of two subspaces, S1 and S2, is dense in the Hilbert space. The conversation also mentions evaluating S2 for density and closeness, and the use of definitions for closed, dense, and spans. Through the conversation, it is determined that S1 + S2 is dense, but not closed, and further assistance may be needed to fully understand the concept.
  • #1
Raven2816
20
0

Homework Statement


i have {ej} is an orthonormal basis on a hilbert space
S1 is the 1-dimensional space of e1 and
S2 is the span of vectors ej + 2e(j+1)

eventually i need to show that S1 + S2 is dense in H and also evaluate
S2 for density and closedness

Homework Equations



i know the def. of closed, dense, spans, etc...

The Attempt at a Solution



well, i know that i need to show that S1+S2 is dense by showing that its closure = my orthornormal basis. i think S2 is closed but not dense, but can an undense set + a dense set be a dense set?
 
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  • #2
e1 is in S1 and 0 is in S2, so e1 = e1 + 0 is in S1 + S2.
Is e2 in S1 + S2?
 
  • #3
yes...e2 = e1 + e1 + 0?
 
  • #4
Raven2816 said:
yes...e2 = e1 + e1 + 0?
No, e1 + e1 is not e2. But (e1 + e2) - e1 is e2.
 
  • #5
then S2 is dense afterall, but not closed. ...at least from what I've worked out since.
 
  • #6
Raven2816 said:
then S2 is dense afterall, but not closed. ...at least from what I've worked out since.
I haven't looked at this carefully, but my first impression is that you are correct. Do you need further help with this?
 

1. What is meant by "Showing S1+S2 is Dense"?

Showing S1+S2 is dense in Hilbert Space means that the linear combination of two subsets of a Hilbert Space, S1 and S2, contains elements that are arbitrarily close to any given element in the Hilbert Space. In other words, the elements of S1+S2 are able to approximate any element in the Hilbert Space with high precision.

2. Why is it important to show that S1+S2 is dense in Hilbert Space?

It is important to show that S1+S2 is dense in Hilbert Space because it ensures that the elements in S1+S2 are able to approximate any element in the Hilbert Space with high precision. This is crucial in many applications, especially in physics and engineering, where precise calculations and measurements are necessary.

3. How is the density of S1+S2 in Hilbert Space proven?

The density of S1+S2 in Hilbert Space can be proven by showing that for any element x in the Hilbert Space, there exists a sequence of elements from S1+S2 that converges to x. This shows that the elements of S1+S2 are able to approximate x with high precision, thus proving the density of S1+S2 in Hilbert Space.

4. What are some examples of subsets S1 and S2 that can be combined to form a dense set in Hilbert Space?

Examples of subsets S1 and S2 that can be combined to form a dense set in Hilbert Space include polynomial functions and trigonometric functions. These sets are commonly used in Fourier analysis and can be used to approximate any continuous function in the Hilbert Space.

5. Can S1+S2 be dense in Hilbert Space if either S1 or S2 is not dense?

No, S1+S2 cannot be dense in Hilbert Space if either S1 or S2 is not dense. In order for S1+S2 to be dense, both S1 and S2 must be dense in Hilbert Space. If one of the subsets is not dense, then the linear combination of the two will also not be able to approximate all elements in the Hilbert Space with high precision.

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