Originally Posted by

**KroneckerDelta** The journal I am reading through contains the following theorem...

Theorem: The set $\{w_n(y,\tau)\}$ is linearly independent and complete in $L_2(E).$

(If you are interested the work is on the heat equation and $w_n$ represents the fundamental solution given at a denumerable number of points and $E$ is the lateral surface of a cylinder under consideration).

I am fine with the proof of linear independence, my question is how do you prove a sequence is complete in functional analysis in general?

In the proof he lets an $f(y,\tau)$ be an arbitrary function in $E$ and

$$\int_{0}^{T}d\tau\int_{S}f(y,\tau)w_n(y,\tau) \,dS = 0, \, n=1,2,3,...$$ Shouldn't that dS be dy?

He then states that if $f(y,\tau) = 0$ then the sequence $\{w_n(y,\tau)\}$ is complete.

Sorry if this is a stupid question above, just like details of why we do the above in the proof.