Inner product Definition and 312 Threads

Let S be a finite set of real numbers. What is a natural inner product to define on the space of all functions f:S->R? I want to approximate an arbitrary function with a polynomial of a fixed degree (both of which are defined only on S), and I want to use projections to do it, but I have no...

I'm getting some confusing information from different sources. If an inner product satisfies conjugate symmetry, it is called Hermitian. But the definition of a hermitian inner product says it must be antilinear in the second slot only. Doesn't conjugate symmetry imply that it's antilinear in...

Hi guys, there's a problem to which me and my pals just can't seem to get an answer that is congruent the answer on the back of the book. The homwork is due in two days and I just want to make sure the book is truly wrong. The question's simple enough: given f,g continuous on the circle...

Hi, I need to use the Cauchy-Schwartz inequality to prove the following inequality. \left( {a_1 + ... + a_n } \right)^2 \le n\left( {a_1 ^2 + ... + a_n ^2 } \right),\forall a_i \in R When does equality hold? The Cauchy-Schwartz inequality is \left| {\left\langle {\mathop...

A question reads: Let V be a vector in an inner product space V show that ||v|| >= ||proj(u) v|| holds for all finitie dimensional subspaces of U. Hint: Pythagorean Theorm. Okay... where on Earth do i begin? I thought perhaps I should expand the RIGHT side of the equation, but...

Oh, its algebra time again! A Question reads: let ||u|| =1 ||v|| = 2 ||w|| = 3^0.5 (or root 3) <u,v>=-1 <u,w>=0 and <v,w>=3 Given this information, who that u + v = w I gave it my best show. i know that ||u|| (im going to write |u| for slimpicity) ... i know that |u|...

Hi I'm stuck on the following question and I have little idea as to how to proceed. Note: I only know how to calculate eigenvalues of a matrix, I don't many applications of them(apart from finding powers of matrices). Also, I will denote the inner product by <a,b> rather than with circular...

Hi, Say I have two inner product spaces, V and W. What is the definition of their tensor product? Is this product naturally always an inner product space? Thank! :smile:

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...