Showing S1+S2 is Dense in Hilbert Space

[Raven2816]

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?

 

[Jimmy Snyder]

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

 

[Raven2816]

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

 

[Jimmy Snyder]

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

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

 

[Raven2816]

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

 

[Jimmy Snyder]

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?