Inner product Definition and 304 Threads

My question is: Let V = \mathbb{R}_1 [x] be the vector space of polynomials in x of degree at most 1. For f(x) \, , \, g(x) \in \mathbb{R}_1 [x], define: <f(x) \, , \, g(x)> \, = \int_0^1 x^2 f(x) g(x) dx Show that this defines an inner product on \mathbb{R}_1[x]. (You may assume...

What is the proof that the inner product of two functions f(x) and g(x) is \int_{a}^{b} f(x)g(x)dx Or is this actually a definition of the inner product for functions? If it is a definition, then what is it based on? Thank you

Im in need of some guidance. No answers, just guidance. :smile: Question. Let (x_m) be a Cauchy sequence in an inner product space, show that \left\{\|x_n\|:n=1,\dots,\infty\right\} is bounded. proof From the definition we know that all convergent sequences are Cauchy...

Question) Define (f|g) = \int_0^2 (1+x^2)f(x)g(x)dx on V = C([0,2],\mathbb{R}). Then this is an inner product space because it obeys the four axioms 1. (f|f) = \int_0^2 (1+x^2)f^2(x)dx \geq 0 since f^2(x) \geq 0 for all x \in [0,2] 2. (f|g) = \int_0^2 (1+x^2)f(x)g(x)dx = \int_0^2...