margaret37 said:Thank you for answering, unfortunately I'm still confused.
Why wouldn't it be closed? If I set up a sequence within S, wouldn't it necessarily converge to something also inside S.
A Hilbert space is a mathematical concept that refers to a complete vector space equipped with an inner product. It is named after the German mathematician David Hilbert.
A closed subset of Hilbert space is one in which every convergent sequence of points in the subset has a limit point that is also in the subset. In other words, the subset contains all of its limit points.
One example of a linear subset of Hilbert space that is not closed is the set of all polynomials on the interval [0,1]. This set is not closed because it does not contain all of its limit points, such as the polynomial x^n for n=0,1,2,... which converges to the function f(x)=0, but f(x) is not a polynomial.
To prove that a subset of Hilbert space is not closed, you can show that there exists a convergent sequence of points in the subset that does not have a limit point in the subset.
Understanding examples of linear subsets of Hilbert space that are not closed is important for understanding the properties and limitations of Hilbert space. It also has applications in fields such as functional analysis, quantum mechanics, and signal processing.