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?
A real function inner product space is a mathematical concept that combines the notions of vector spaces and inner products. It is a vector space that is equipped with an inner product, which is a way to measure the angle between two vectors and the length of a vector.
A real function inner product space must satisfy the following properties: linearity, positive definiteness, symmetry, and additivity. Linearity means that the inner product must be distributive and linear with respect to scalar multiplication. Positive definiteness requires that the inner product of a vector with itself is always positive. Symmetry means that the inner product of two vectors is the same regardless of the order in which they are multiplied. Additivity means that the inner product of two vectors added together is the same as the sum of their individual inner products.
Real function inner product spaces have many applications in mathematics and physics. They are used to define norms and distances, which are important tools in studying convergence and continuity of functions. They also play a key role in the development of Fourier series and other types of function expansions. In physics, inner product spaces are used to describe quantum states and to calculate probabilities of measurement outcomes.
The main difference between a real function inner product space and a complex function inner product space is the type of numbers that are used for the inner product. In a real function inner product space, the inner product is defined using real numbers, while in a complex function inner product space, the inner product is defined using complex numbers. This allows for a more general and versatile framework in the complex case, which is why it is often used in more advanced mathematical and physical applications.
Yes, there are some limitations and restrictions on real function inner product spaces. One important limitation is that the inner product space must be finite-dimensional. This means that there is a limit to the number of vectors that can be used to span the space. Additionally, some properties of inner products, such as symmetry, may not hold in certain non-Euclidean spaces. In these cases, alternative definitions of inner products may be used to overcome these limitations.