While reading wikipedia I found the following quote:(adsbygoogle = window.adsbygoogle || []).push({});

"...When this basis is indexed by a discrete (finite or countable) set, the vector representation is just the column of numbers familiar to high-school students.Vector bases may also be continuously indexed.When a quantum mechanical state vector is represented with respect to such a continuous basis it is called a wavefunction."

I know that thebasis of a vector spaceis the set of vectors that spans a space.

I think thatindexing a vector by a discrete setmeans providing a coefficient for each spanning vector, iff the discrete set has the same number of elements as the number of elements in the spanning set. This is where we get the finite nature of the discrete set, i.e. the number of elements is finite, as is the dimension of the vector space, which equals the number of spanning vectors.

Is this correct? If not, what is the correct interpretation?

Finally, what is a"continuously indexed basis"?Is this an infinite dimensional vector--i.e. a vector in R^n, or Hilbert space? (Wouldn't that rather be an infinitely indexed basis?) Or is this a vector space where a vector is defined by a single continuous function, rather than a series of coordinates? i.e. where you have instead of discrete dimensions, a continuum of dimensions--i.e. instead of dimension 2, 3, 4, you might have dimension 1, 1.75, Pi, and so on and everything in between continuously. I really don't understand what is meant by a continuously indexed basis. What is it?

-Mark

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# Def.: Continuously Indexed Vector Basis?

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