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 :
Jesus Christ and His Teachings
Way of enlightenment, wisdom, and understanding
America, a corrupt, depraved, shameless country
On integrity and the lack of it
The test of a person's Christianity is what he is
Ninety five percent of the problems that most people have come from personal foolishness
Liberalism, socialism and the modern welfare state
The desire to harm, a motivation for conduct
On Self-sufficient Country Living, Homesteading
Topically Arranged Proverbs, Precepts, Quotations. Common Sayings. Poor Richard's Almanac.
Theory on the Formation of Character
People are like radio tuners --- they pick out and listen to one wavelength and ignore the rest
Cause of Character Traits --- According to Aristotle
We are what we eat --- living under the discipline of a diet
Avoiding problems and trouble in life
Role of habit in formation of character
Personal attributes of the true Christian
What determines a person's character?
Love of God and love of virtue are closely united
Intellectual disparities among people and the power in good habits
Tools of Satan. Tactics and Tricks used by the Devil.
The Natural Way -- The Unnatural Way
Wisdom, Reason and Virtue are closely related
Knowledge is one thing, wisdom is another
My views on Christianity in America
The most important thing in life is understanding
We are all examples --- for good or for bad
Television --- spiritual poison
The Prime Mover that decides "What We Are"
Where do our outlooks, attitudes and values come from?
Sin is serious business. The punishment for it is real. Hell is real.
Self-imposed discipline and regimentation
Achieving happiness in life --- a matter of the right strategies
Self-control, self-restraint, self-discipline basic to so much in life