Inner product Definition and 304 Threads

Homework Statement I have posted simular questions a couple of times but now I feel I have a better understanding(hopefully). Given a Vectorspace M which is defined as a sequence of realnumber \{r_n\} and where \sum_{r=1}^{\infty} r_n < \infty Show that M has an innerproduct given by...

Hi, I'm interested in constructive QFT and I'd like to pose a question about construction of the one-particle Hilbert space of states for bosons. For fermions satisfying the Dirac equation, the inner product is \langle \psi, \phi \rangle = \int d^3 x \; \psi^\dagger(x) \phi(x). The...

Homework Statement Hi all. The inner product between two functions f(x) and g(x) is defined as: <f | g> = \int f^*(x)g(x) dx, where the * denotes the complex conjugate. Now if my functions f and g are functions of r, theta and phi (i.e. they are written in spherical coordinates), is the...

my question is: if a.b=c with a=any vector b=any vector .=inner product c=resulting scalar is there a way to describe a=f(b,c)? Thanks

Dear All: Any idea for the following interesting question: As we know we can calculate inner product of two wave functions A and B as <A|B>. here both A and B are vector in hilbert space. here we may use fourier transform to get momentum representation of A and B, and get same...

I am working my lonely way through Spivack's "Calculus on Manifolds." (not a registered student anywhere, alas). On p. 23 is a set of problems involving the inner product. I believe I've got it up to d), which asks for a function f:R->R s.t. f is differentiable but |f| is not...

Hi everyone, I need a clarification:I read in E. Butkov's book that an inner product may always be imposed on a finite dimensional linear vector space in a variety of ways...Butkov does not explain the point...Can anyone please clarify this? I wonder what it would be for an...

My apologies in advance for asking what (to me) looks like an extremely stupid question, but I just can't figure it out. 1. Homework Statement : Where is this an inner product: \int_{a}^{b}f(x)g(x) dx a) on C[a,b]? b) on C(R)? The answer is that it is an inner product on a), but not...

I want to consider the space of NxN real matrices as a vector space in which any given NxN matrix can be given as a real weighted sum of at most N^2 basis matrices. I already know how this works if I assume the form for the inner product (eg. (1/N)Tr(matrix product)). However, here's the...

I have some questions about inner product sapces. 1. If V is a vector space over R and ( , ):VxV-->R is an inner product on V, then for v,w in V, is the value of (v,w) independent of my choice of basis for V used to compute (v,w)? 2. If V is an arbitrary n dimensional vector space over...

I have been searching for a way to relate known concepts (known to me) to the computation of the dot product in an effort to understand why it takes the form it does. I ran into a little snippet in a classical dynamics book that seems like it just may be the ticket. Here is what it says...

To find dual basis from the inner product Matrix!? Homework Statement WE know the inner product matrix (capital)Gamma and that's all. How do we "construct" a dual basis? Homework Equations The Attempt at a Solution I know that the orthonormal basis is nothing but a dual...

Homework Statement Prove that \left\langle\alpha x,y\right\rangle-\alpha\left\langle x,y\right\rangle=0 for \alpha=i where \left\langle x,y\right\rangle=\frac{1}{4}\left\{\left\|x+y\right\|^{2}-\left\|x-y\right\|^{2}+i\left\|x+iy\right\|^{2}-i\left\|x-iy\right\|^{2}\right\}...

Homework Statement Prove that the normed linear space l_{\infty}^{2} is not an inner product space. Homework Equations parallelogram law; \left\|x+y\right\|^2+\left\|x-y\right\|^2=2\left\|x\right\|^2+2\left\|y\right\|^2 The Attempt at a Solution Well, I tried to apply...

Homework Statement Not all normed linear spaces are inner product spaces. Give examples. Homework Equations all equations and conditions constructing inner and normed linear spaces. The Attempt at a Solution Well, I tried some of spaces like L space, but I didn't find any logical...

Can someone explain the additivity property of inner product spaces to me please

Homework Statement The following is from the book Linear Algebra 3rd Edn by Stephen Friedberg, et al: Here aj are scalars of field F and vj are vectors of inner product space V. Homework Equations Theorem 6.3: The Attempt at a Solution Now I don't understand why theorem 6.3...

Homework Statement Let V be an inner product space. Then for x,y,z \in V and c\inF, where F is a field denoting either R or C, prove that <x,x> = 0 if and only if x=0. Notes on notation: Here <x,y> denotes the inner product of vectors x and y on some vector space V. Homework...

Homework Statement Let H be a Hilbert space. Prove that if \left\{ x _{n} \right\} is a sequence such that lim_{n\rightarrow\infty}\left\langle x_{n},y\right\rangle exists for all y\in H, then there exists x\in H such that lim_{n\rightarrow\infty} \left\langle x_{n},y\right\rangle =...

My differential forms book (Flanders/Dover) defines an inner product on wedge products for vectors that have a defined inner product, and uses that to define the hodge dual. That wedge inner product definition was a determinant of inner products. I don't actually have that book on me right...

Homework Statement Construct an orthogonal basis of R2 equipped with the non-standard inner product defined for all X, Y belonging to R2, by <X,Y> = X^T AY with A = 2 1 1 3 The Attempt at a Solution So it seems pretty trivial, but I can't seem to get the answer. So my approach is 1)...

Hello, I was trying to understand Green's function and I stumbled across the following statements which is confusing to me. I was referring to the following site http://www.math.ohio-state.edu/~gerlach/math/BVtypset/node79.html Here the author says the following "What if $ u$ is...

In Quantum mechanics books they usually first introduce a vector space called the ket-space and then associate using (Riesz representation theorem I believe) to each ket a corresponding element of the linear dual space. Then they write the inner product of |x\rangle and |y\rangle (say) by...

Can somebody give me an other metric space that is not dependent on the inner product i mean which is not derived from the inner product between two vectors.

Dear all: I have a problem about the inner product of a function. Give a function \begin{displaymath} f(x) = \left\{ \begin{array}{ll} x & \textrm{if $x \in [0,1]$}\\ -x+2 & \textrm{if $x \in (1, 2]$} \end{array} \end{displaymath} \{ What's the value of the inner product of...

I come over this in my coding theory but can't understand it. It says finite fields do not fulfil the definition of inner product space like other fields (R and C. Why? How is the proof? Thanks!

let V be a vector space with inner product, and T:V->V linear trans. then for V on R, prove that for every v in V, <v,T(v)>=0 iff T*=-T. now i got so far that: from <v,T(v)>=0 we have <v,(T+T*)(v)>=0 for every v, here I am stuck, i guess if it's for every v, if i were to write (T+T*)(v)=av...

Homework Statement Given the Vectorspace V of the real polynoms and the sub space L(1, t, t^2). On V there's a inner product defined as follows: <u(t), w(t)> = integral(u(t)*w(t), dt, -3, 3) I have to find the inner product of the subspace in reference of the basis (1, t, t^2)...

Does it surprise you that the fundalmentals of wave mechanics fits so nicely into an inner product space. I assume this kind of algebra existed long ago but QM seem to fit perfectly into it. How amazing is that?

Hello all, I have two questions that are fairly general, but slightly hazy to me still. o:) 1) Can we consider the inner product to be a bilinear function, or not? I would like to think of it as a mapping from an ordered pair of vectors of some vector space V (i.e. VxV) to the field (F)...

Homework Statement What s the inner product <2011|0011> Homework Equations C_{m_1m_2}=<l_1l_2m_1m_2|lml_1l_2> The Attempt at a Solution I'm not sure how to exactly solve this question. The first thing that came to my mind was the Clebsch-Gordan equation, since that's what it looks like...

Hello, I am working on an assignment were I have shown that a certain equation defines an inner product, which was simple enough. Te equation was: \left\langle {f,g} \right\rangle = \int\limits_0^1 {f\left( x \right)g\left( x \right)x^2 dx} My question then is: How do i state an equation for...

The question requires me to check whether the following formulae satisfy the properties of an inner product given the linear space of all real polynomials. f(1)g(1) \left(\int_{0}^{1}f(t)dt\right)\left(\int_{0}^{1}g(t)dt\right) The properties are satisfied in both cases (at least, that's my...

Assume that m<n and l_1,l_2,...,l_m are linear functionals on an n-dimensional vector space X . Prove there exists a nonzero vector x \epsilon X such that < x,l_j >=0 for 1 \leq j \leq m. What does this say about the solution of systems of linear equations?This implies l_j(x)...

1) C[-1,2] is a space of all continues functions f: [-1,2] -> C (complex) Is: <f,g> = \int_{-1}^{2}|f(t) + g(t)|dt an inner product of C[-1,2]? I think that the answer is no because: <f+g, h> \neq <f,h> + <g,h> for some f and g. this can happen when all the functions are positive...

Hey guys, I am studying atm and looking at this book: "Introduction to Hilbert Space" by N.Young. For those who have the book, I am referring to pg 32, theorem 4.4. Theorem If x1,...,xn is an orthogonal system in an inner product space then, ||Sum(j=1 to n) xj ||^2 = Sum(j=1 to...

If α,β,γ are vectors in the Euclid space V, please show that |α-β||γ|≤|α-γ||β|+|β-γ||α|,where |α|=√(α,α) and point out when the equal mark holds. Can someone help me out?

Very basic question but could someone briefly explain why the inner product for complex vector space involves the conjugate of the second vector. Of course if imaginary component is 0 then this reduces to dot product in real vector space. And I see that this definition makes sense to calculate...

Definition: Let f:V->V be a linear transformation on an inner product space V. The adjoint f* of f is a linear transformation f*:V->V satisfying <f(v),w>=<v,f*(w)> for all v,w in V. My question is would <f*(v),w>=<v,f(w)> be equivalent to the above formula in the definition? If so why...

Is there any difference between an inner product and a dot product?

"Suppose V is a (real or complex) inner product space, and that T:V\rightarrow V is self adjoint. Suppose that there is a vector v with ||v||=1, a scalar \lambda\in F and a real \epsilon >0 such that ||T(v)-\lambda v||<\epsilon. Show that T has an eigenvalue \lambda ' such that |\lambda...

Hi, I'm looking at the definition of the inner product of two vectors in \mathbb{C}^n . One source is talking about how the definition of an inner product must be modified to account for vectors with complex components and says: He then goes on to say that we can rewrite conjugate...

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: