Homework Help: Hilbert spaces

1. May 10, 2007

Raven2816

1. The problem statement, all variables and given/known data
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

2. Relevant equations

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

3. 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?

2. May 10, 2007

Jimmy Snyder

e1 is in S1 and 0 is in S2, so e1 = e1 + 0 is in S1 + S2.
Is e2 in S1 + S2?

3. May 10, 2007

Raven2816

yes...e2 = e1 + e1 + 0?

4. May 10, 2007

Jimmy Snyder

No, e1 + e1 is not e2. But (e1 + e2) - e1 is e2.

5. May 10, 2007

Raven2816

then S2 is dense afterall, but not closed. ...at least from what i've worked out since.

6. May 10, 2007

Jimmy Snyder

I haven't looked at this carefully, but my first impression is that you are correct. Do you need further help with this?