such that for every element a of A, the net (or sequence)
has limit a.
Similarly, a left approximate identity is a net
such that for every element a of A, the net (or sequence)
has limit a.
An approximate identity is a right approximate identity which is also a left approximate identity.
Every C*-algebraA has an approximate identity of positive elements of norm ≤ 1; indeed, the net of all positive elements of norm ≤ 1; in A with its natural order always suffices.
An approximate identity in a convolution algebra plays the same role as a sequence of function approximations to the Dirac delta function (which is the identity element for convolution). For example the Fejér kernels of Fourier series theory give rise to an approximate identity.
The first two identities are known as the double angle identities, while the last is called the Pythagorean identity, since it also can be derived from the Pythagorean theorem (using our geometric description of sin and cos).
This approximation is also evident in the geometric definition of sin, since small arcs are relatively straight, so that the vertical coordinate is approximately equal to the distance along the circle.
While these identities prove to be useful in a number of advanced mathematics courses, they are surprising and beautiful in their own right.
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