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...
I have an assignment question here using calculus of variations.
Question:
Show that the length ℓ of a curve on the surface of a sphere of radius a is
ℓ=int sqrt(theta_dot^2+(sin(theta))^2*phi_dot^2) dt
Hence show that the shortest distance between two points on the sphere is given by part...