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## Homework Statement

If there exists no function, f(x), except zero, with the property that

[tex]\int_{a}^{b}{\phi_{n}(x)}f(x)w(x)dx=0[/tex]

for all [tex]\phi_{n},[/tex] then the set {[tex]\phi_{n}(x)[/tex]} is said to be complete.

Write a similar statement expressing the completeness of a set of basis vectors in three-dimensional Euclidean space. Explain in simple terms what it means for vectors.

## Homework Equations

w(x) is the positive real weight function (I do not know what this means)

I'm planning on involving the inner and/or outer product here.

## The Attempt at a Solution

I do not have a solid understanding of the equation above. Here is what I do know:

I know that in 3D Euclidean space the basis vectors are:

[tex]\hat{x}= (1,0,0)\qquad\hat{y}=(0,1,0)\qquad\hat{z}=(0,0,1)[/tex]

I am also familiar with their inner and outer product results. I know that in 3-D it is possible to have a vector(non-zero) whose cross or dot product with 1 of the basis vectors yields 0. I know of only 1 vector(the null vector), in 3-D, whose cross or dot product with all of the basis vectors, in turn, equals 0.

I think I can answer this problem if I know what [tex]\phi_{n}[/tex] represents.

In the case of the vectors, does it represent [tex]\hat{x},\hat{y},{and},\hat{z}[/tex]?

Or does it represent only 1 of the base vectors? Since I have said that I can think of a vector that will yield 0 crossed or dotted with 1 base vector, then I think [tex]\phi_{n}[/tex] would have to mean all of the basis vectors. If that is the case, am I thinking about 3 different integrals or am I thinking about 1 integral with 5 terms being multiplied?

I think it has to be 3 different integrals but I do not understand how the notation expresses that. I am familiar with the summation notation [tex]\hat{x}_{i}[/tex]. Is that what we are talking about here?

Also, are we talking about the inner product, outer product or both?

One last question unrelated to this problem. I am not very confident in my Latex ability and I often preview my posts many times. Every time I do this, the original structure of the post (The problem statement, relevant Eqs, and solution attempt) get added each time I preview my post. I have to delete the repeats each post. Does anyone know a way around this?

Sorry for rambling on. Thank you for taking the time to read/respond to my questions.