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How is

$$\langle f, g\rangle \text{where} f(x,y), g(x,y)$$

defined, where

$\langle \cdot ,\cdot \rangle$ means the inner product of $f$ and $g$?

(I primarily searched How can we define the Inner Product of multi-variable functions?)

KYHSGeekCode
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1 Answers1

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Inner product for functions (one variable or many variables) is usually defined through definite integrals. One assumes all the functions have the same domain. Take an interval (or a region) in the domain of the function. The inner product is the definite integral over that interval/region of the product of those two functions concerned.

P Vanchinathan
  • 19,504
  • You mean $$\idotsint_D f(x_1,x_2,x_3,\cdots,x_n) \overline {g(x_1,x_2,x_3,\cdots,x_n)} dx_1dx_2\cdots dx_n$$? – KYHSGeekCode Sep 10 '18 at 09:05
  • Yes, in case you are dealing with complex-valued functions take complex conjugates. – P Vanchinathan Sep 10 '18 at 09:06
  • Thanks, then can we define orthogonality by saying that expression is 0? – KYHSGeekCode Sep 10 '18 at 09:13
  • All the usual things are available like orthogonality: inner product being zero. – P Vanchinathan Sep 10 '18 at 09:26
  • @PVanchinathan For vectors the inner product is defined as a sum. Why we need to integrate? What is the purpose of $dx$? – user599310 May 12 '20 at 17:15
  • A function $f(x)$ here is a specification of values for every $x$ in some interval (or domain set $S$. A vector of $n$ components $v=(x_1,x_2,\ldots, x_n)$ can also be regarded as function, but on the finite set ${1,2,\ldots, n}$, the value at $i$ being $x_i$ for $i=1,2,...n$. As vectors involve only a finite number of numbers we can use summing them to use up ALL the information contained in them to get inner products. But functions being an uncountable set of number or pairs of numbers $(x, f(x))$ one cannot sum them. Integration fits the bill. ( Riemann integral is limit of a sum) – P Vanchinathan May 13 '20 at 03:20