Question

In the context of Hilbert spaces, the term "separable" means that the space:

a) Can be easily separated into subspaces
b) Can be densely populated
c) Has a countable dense subset
d) Is only accessible to a specific group

Answer 1

In the context of Hilbert spaces, the term "separable" means that the space has a countable dense subset, which allows any point in the space to be approximated by a sequence from this subset.

Step-by-step explanation:

In the context of Hilbert spaces, the term "separable" refers to the property of having a countable dense subset. A Hilbert space is said to be separable if and only if there exists a countable set within the space such that its closure is the entire space, meaning that any point in the Hilbert space can be approximated arbitrarily well by a sequence of points from this set.

This is crucial for applications in mathematics and physics because it ensures that the space is not "too large" and allows for a more workable structure when dealing with infinite dimensions.