complete set

eugene · Yang-Mills Theory Offline Joined: 16 May 2004 Posts: 970

Prove that any finite set couldn't be complete in $L_2$ (I don't know whether the word "complete" is acceptable)

**Posted:** Thu Feb 10, 2005 5:13 am

Kent Merryfield

One question: $L^2$ of what? Of a bounded inteval? Of $\mathbb{R}$ ? Of the integers? (The answer is the same in all of those cases, but you probably should be specific.)

What I assume you mean by "complete" is this: set of elements of an inner product space is complete if and only if its span is dense in that inner product space.

Equivalent statement: $W$ is complete iff $(\text{span}(W))^{\perp}=\{0\}.$

It boils down to $L^2$ being an infinite dimensional vector space.

**Posted:** Thu Feb 10, 2005 9:16 am

eugene · Yang-Mills Theory Offline Joined: 16 May 2004 Posts: 970

I'm sorry for being not correct: by $L_2$ i meant the set of all square-summable funtions on $[a,b]$ , i.e. the set of all measurable fucntions s.t. $\int_{a}^{b}f^2(x)dx<\infty$
In my meaning "complete" set of fucntions means that the only elemnt, which is orthogonal to any $f\in{L_2}$ is $0$

**Posted:** Thu Feb 10, 2005 9:56 am

Kent Merryfield

So, $L^2$ of a bounded interval it is. And I assume in that last sentence you mean that the only element orthogonal to any $f$ in the set in question (not all of $L^2$ ) is zero. Assuming that, your definition of "complete" is the same as mine.

To actually prove that no finite set is complete, it suffices to exhibit an infinite orthogonal set. Let $f_n(x)=\sin\left(\frac{n(x-a)}{b-a}\right)$ for $n\in\mathbb{N}.$ This is an infinite orthogonal set. (Divide them all by the right constant and it's an infinite orthonormal set.) Now suppose $W$ is any finite set. Thus the span of $W$ is a finite-dimensional subspace. Since $\{f_n\}$ , being orthogonal, is independent, there must be some $n$ for which $f_n\not\in\text{span}(W).$ (You can't have infinitely many independent vectors in a finite-dimensional vector space.) We can explictly construct $g\in\text{span}(W)$ such that $f_n-g$ is orthogonal to everything in $\text{span}(W).$ (The construction is a standard part of the Gram-Schmidt algorithm.) That proves that $W$ is not complete.

**Posted:** Thu Feb 10, 2005 10:18 am