waddles said:Homework Statement
Let S be a linearly independent subset of a Hilbert space. Prove that span(S) is a subspace, that is a linear manifold and a closed set, if and only if S is finite.
Homework Equations
The Attempt at a Solution
Assuming S is finite means that S is a closed set (a finite subset of a metric space is closed). I think that this will help to prove span(S) is a closed set but I am a bit stuck.
A Hilbert space is a mathematical concept that refers to a vector space with an inner product operation defined on it. It is a generalization of Euclidean space and is often used in functional analysis and quantum mechanics.
A subset of a Hilbert space is said to be linearly independent if none of its elements can be expressed as a linear combination of the others. In other words, no element in the subset is redundant and each one contributes uniquely to the span of the subset.
The span of a linearly independent subset of a Hilbert space is the set of all possible linear combinations of the elements in the subset. In other words, it is the set of all vectors that can be created by scaling and adding the elements in the subset.
The span of a linearly independent subset of a Hilbert space is a subspace because it satisfies the three defining properties of a subspace: closure under addition, closure under scalar multiplication, and containing the zero vector. Since the subset is linearly independent, any combination of its elements will not create any new vectors outside of the span, making it a closed set.
The span of a linearly independent subset of a Hilbert space is finite because a Hilbert space is a finite-dimensional vector space. This means that the span will only contain a finite number of linearly independent elements, as the dimension of the vector space determines the maximum number of linearly independent vectors it can contain.