A closed subspace is a subset of a vector space that contains all of its limit points. In other words, any sequence of points in the subspace that converges must also have its limit point in the subspace.
The sum of two closed subspaces is defined as the set of all possible sums of vectors from each subspace. In other words, if A and B are two closed subspaces, then their sum is denoted as A + B and is defined as {a + b | a ∈ A, b ∈ B}.
Yes, it is possible for the sum of two closed subspaces to be closed. This happens when the two subspaces have a non-empty intersection, meaning there are vectors that belong to both subspaces. In this case, the sum of the subspaces will also be a closed subspace.
One example is the subspaces A = {(x, y) | x ≥ 0, y = 0} and B = {(x, y) | x = 0, y ≥ 0} in R^2. The sum of these two subspaces is the x-axis, which is not a closed subspace. Another example is the subspaces A = {(x, y) | x ≥ 0, y = 0} and B = {(x, y) | x = 0, y > 0} in R^2. The sum of these two subspaces is the first quadrant, which is also not a closed subspace.
The fact that a sum of two closed subspaces can sometimes not be closed has important implications in functional analysis and topology. It means that in general, the sum of two closed subspaces is not always a closed subspace, and thus cannot be treated as such in mathematical proofs and calculations. This concept is also important in understanding the properties of vector spaces and their subsets.