Thread: Completeness

I was looking at the following proof that the quotient space X/M where X is a Banach space and M is a closed subspace:

PlanetMath: quotients of Banach spaces by closed subspaces are Banach spaces under the quotient norm

Can someone elaborate for me on the second to last line beginning x-s_k+M? In particular I'm shaky on the last two equalities.

Thanks.

In a quotient space, the coset containing x+y is (by definition) the sum of the coset containing x and the coset containing y. In other words, (x+y) + M = (x+M) + (y+M). By induction, this extends to any finite sum of cosets.

That is all that is happening in the line The first two equalities are using that fact about sums of cosets (remember that ), and the last equality comes from the definition of .