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Most textbooks say, that given a (countable) basis ${|\phi_n\rangle}$ of a Hilbert space, that every vector $|\psi\rangle$ of the space can be written as:

$$\psi\rangle=\sum_{n=1}^\infty a_n|\phi_n\rangle$$

But this an infinite linear combination, and every vector should be expressed by a finite linear combination.

So I think that they are an orthogonal set, which generates the vector space as an infinite linear combination (the difference between infinite and finite LC means that orthonormal sets are usually smaller than basis).

But for the rest of the properties, ${|\phi_n\rangle}$ looks like a basis to me.

So, is ${|\phi_n\rangle}$ a basis or just an orthogonal set?

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What is your question? – Niel de Beaudrap Dec 5 '13 at 9:21
@Niel de Beaudrap Updated. – jinawee Dec 5 '13 at 9:26
in qm what is being called "basis" is indeed a complete orthogonal set. I think it is called basis because it resembles of the concept of basis in linear algebra, not the concept of basis of a metric (or more generally, topological) space. – Jia Yiyang Dec 5 '13 at 9:33
up vote 4 down vote accepted

It is a basis. The trouble is that "basis" is actually vague if removed from specific context, and what we have here is a Schauder basis. The definition you learned in basic linear algebra that requires a finite linear combination is a Hamel basis.

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So Hermite, Legendre... polynomials form a Schauder basis? – jinawee Dec 5 '13 at 9:31
That's right: the Hermite polynomials form an orthogonal Schauder basis if the inner product has an appropriate (Gaussian) weight function, similarly for Legendre with different caveats, but yes. Another example is the Fourier basis $\{1,\cos nx,\sin nx:\;n>0\}$ for (equivalence classes of) functions of period $2\pi$. – Stan Liou Dec 5 '13 at 9:46

It's a matter of definition.

According to Hirzebruch/Scharlau, Einführung in die Funktionalanalysis (1971), Definition 21.10:

An orthonormal system $\{x_i\}_{i\in I}$ that meets the [proven equivalent] conditions of this theorem is called a Hilbert basis or just basis of $X$.

Wikipedia calls it an orthonormal basis, which is a Schauder basis if your infinite-dimensional Hilbert space is separable.

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