Assuming that you are, in fact, talking about an infinite dimensional vector space and that

is an orthonormal vector space, then those are just extensions of the corresponding notions for finite dimensional vector spaces.

In particular, if

, we can determine each of the coefficients,

, by taking the inner product of u with

:

. But

if i= j, 0 otherwise so that sum collapses down to the single ualue,

. That's where line (2) comes from: Since

,

.

The norm, in any innerproduct space, is defined by

so if

, then

. But since, again, since

are orthonormal,

= 1 if i= j, 0 otherwise, so that sum reduces to

which, as above, is exactly the same as

.