# When a set of functions becomes complete?

I know that a set of functions are said to form a complete basis on an inteval if any function on that interval can be expressed as a linear combination of the functions in the set. I also know that every function in the set are orthogonal.

Now what is what is the condition(s) that a set of functions has to statisfy to become complete?

That is, how to prove a set of orthogonal functions span a space?

• I don't understand the question. I think you are meaning to ask when an arbitrary sequence of orthogonal vectors in some inner product space is complete, which is when the set spans the space.
– JMJ
Commented May 27, 2016 at 23:00
• @ALB yes, I'm asking how to prove that they span the space? Commented May 27, 2016 at 23:27
• This is the same as any other proof a set of vectors spans a space, which is to show that an arbitrary element of the space can be written as a linear combination of the basis. In the finite-dimensional case this can be done using a matrix representation and in the infinite-dimensional case you need to prove that the partial sums of the orthogonal set converge to the point.
– JMJ
Commented May 27, 2016 at 23:36

A metric space is said to be "complete" if every Cauchy sequence converges.

For example: Let $(X, \mu)$ be a measure space. Then $L^P(X)$ is complete under the $L^P$ norm, for $p \in [1,\infty]$. [It is a Banach space.]

Every finite dimensional normed vector space is also complete. (This this can be explained by the Lipschitz equivalence to the euclidian norm.)

Notions of completeness need not be restricted to a set of functions. For example, $\mathbb{R}$ is complete, since every Cauchy sequence $\{a_n\}$ converges in $\mathbb{R}$.

In fact, it can be proven that $\mathbb{R}$ is the completion of $\mathbb{Q}$; i.e.: take a sequence of rationals that is Cauchy and define it's limit to be a real number.

• I do not think OP was asking about metric completeness, since this interpretation does not make sense in the context of his question. He is asking about the completeness of a set of orthogonal functions--e.g. whether or not the set spans the space.
– JMJ
Commented May 27, 2016 at 23:20
• Oh, in that case, just when they span the space. Commented May 27, 2016 at 23:21
• @AndresMejia and how to prove such a thing? Commented May 27, 2016 at 23:28
• math.stackexchange.com/questions/268471/… Commented May 27, 2016 at 23:30
• @OmarNagib can you give the definition of complete that you are using? Commented May 27, 2016 at 23:31