Thread: Orthogonal Complement

What is Y^{\perp} if Y= span{e_1,e_2, \cdots , e_n} \subset \mathbb{l}^2, where e_j=(\delta_{ij}) ?

e_1 = (1,0,0,0, \ldots)
e_2 = (0,1,0,0, \ldots) etc.

It depends: Is your a fixed integer or not? If it's not, then the orthogonal complement is because is dense in . If, on the other hand, is fixed, then is a finite dimensional subspace of and so is closed, therefore you have . (It should be clear from here who is in this last case, since to represent every element of you need all the elements in your sequence wich are greater or equal than the given ).

Hi Jose.

Thank you for your message! Very helpful man.
So in the second case assuming n is fixed at say n=3.
How would I write the Y^{\perp} in set notation?

say x = (\xi_j) \in l^2 is it then:

Y^{\perp} = \{ x = (\xi_j) \vert \sum^{j=4}^{\infty} < \infty \} ?

or should I express interms of e_n 's.

For example you have and you want to express a sequence in , then what you are missing is the terms , so that should be all sequences of the form , and a basis for should be