The discussion centers around the characterization of inner product spaces, specifically regarding the vector space of real functions defined on a closed interval [a,b] and the implications of using the function ##1/x## within this context. Participants explore the conditions under which the inner product is defined and the integrability of functions.
Participants express disagreement regarding the applicability of the inner product definition to the function ##1/x##, with no consensus reached on whether Wolfram's statement is entirely accurate.
There is an emphasis on the need for functions to be integrable or continuous for the inner product to be well-defined, highlighting limitations in the generality of the statement made by Wolfram.
Why shouldn't it converge? ##\int_a^b \dfrac{1}{x}\dfrac{1}{x}\,dx = -\dfrac{1}{b}+\dfrac{1}{a}\,.##Mr Davis 97 said:Wolfram says that an example of an inner product space is the vector space of real functions whose domain is an closed interval [a,b] with inner product ##\langle f, g\rangle = \int_a^b f(x) g(x) dx##. But ##1/x## is a real function, and ##\langle 1/x, 1/x\rangle## does not converge... So how is this an inner product space?
Sorry, I meant to replace ##a## with ##-1## and ##b## with ##1##fresh_42 said:Why shouldn't it converge? ##\int_a^b \dfrac{1}{x}\dfrac{1}{x}\,dx = -\dfrac{1}{b}+\dfrac{1}{a}\,.##
But the function ##x \mapsto \dfrac{1}{x}## you mentioned isn't defined on ##[-1,1]##. You also need functions which are at least integrable, usually Lebesgue integrable, or continuous. Real valued alone is too weak, because at least the inner product must be defined!Mr Davis 97 said:Sorry, I meant to replace ##a## with ##-1## and ##b## with ##1##
So is what Wolfram said incorrect?fresh_42 said:But the function ##x \mapsto \dfrac{1}{x}## you mentioned isn't defined on ##[-1,1]##. You also need functions which are at least integrable, usually Lebesgue integrable, or continuous. Real valued alone is too weak, because at least the inner product must be defined!
Do you have a link?Mr Davis 97 said:So is what Wolfram said incorrect?