Website owner: James Miller
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 :
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