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EXAMPLES OF INNER PRODUCTS DEFINED ON VARIOUS ABSTRACT SPACES

Examples of inner products defined on various abstract spaces:

1. Space Rn of all n-tuples of real numbers. The usual inner product defined on this space is the dot product

where and .

2. Space Cn of all n-tuples of complex numbers. The usual inner product defined on Cn is the dot product

where u = (u1, u2, .... , un) and v = (v1, v2, .... , vn).

3. Vector space of mxn matrices over R. The following is an inner product on this space:

(A,B) = tr (BTA)

where tr stands for trace, the sum of the diagonal elements.

4. Vector space of mxn matrices over C. The following is an inner product on this space:

(A,B) = tr (B*A)

where B* denotes the conjugate transpose of the matrix B.

5. Vector space of real continuous functions on the interval a t b. An inner product on this space is given by:

6. Vector space of complex continuous functions on the (real) interval

a t b. An inner product on this space is given by:

7. l2 - space (or Hilbert space) consisting of all infinite sequences of real numbers (a1, a2, .... , an , ...) satisfying

An inner product on this space is given by :