Let has compact support} and f has compact support, Show that the closures of F1 and F2 with respect to the norm are the same. If H1 is the completion of functions on Rd with compact support,...

Let

has compact support} and

f has compact support,

Show that the closures of F₁
and F₂
with respect to the norm

are the same.

If H₁
is the completion of

functions on R_d
with compact support, relative to the norm given by (38.11), show H₁
is a Hilbert space.