I couldn’t remember how to construct an example of a two closed subspaces of a Banach space such that their sum is not closed, so I searched online. I found a discussion of an example at physicsforum, and a paper by Schochetman, Smith, and Tsui that characterizes when the sum of two closed subspaces of a Hilbert space will be closed.
A “naturally occurring” example of close subspaces of Banach spaces not having closed sum is H¹(T) and it’s conjugate as subspaces of L¹(T), Equally $latex H^infty(T)$ and it’s conjugate or the disc algebra and it’s conjugate work in $latex L^infty(T)$.
l^1 is the conjugate of c_0, and l^infty is the conjugate of l^1.
frac{(x-x1) + i(y1 as x goes to x1 and y goes to y2
There is a theorem by Kato that M+N is closed iff the sum of their annihilators is the annihilator of their intersection, iff M+N is the preannihilator of the intersection of their annihilators, ie using \perp as notation for annihilators,
M+N is closed iff M\perp + N\perp = (M \cap N)\perp
iff M + N = \perp(M\perp cap N\perp)
A proof can be found in Mennicken and Sagraloff, “On Banach’s closed range theorem” Arch. Math. 33(1980) 461-465