An inner product space is a vector space equipped with an inner product. The inner product is a generalisation of the "dot" product often used in vector calculus.

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Inner Product of Imaginary Component and Real Component of an Orthogonal Function

I would like to evaluate the inner product of two components of a basis function that is part of a complete set. The two components are the imaginary and real components of the basis function. Could I ...
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Proportinal line elements imply preservations of angles.

Consider a Riemannian manifold $(M,g)$, and a variation of the line element $\delta ds^2$ that is proportional to the original line element $ds^2$. This is $\delta ds^2=c ds^2$ for some constant $c$. ...
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Is the orthogonality between Associated Legendre polynomials preserved on an interval [-a,a]

So I am aware of the orthogonality between the Associated Legendre polynomials on the interval $[-1,1]$, that is: \begin{equation} \int_{-1}^{1}P^m_kP^m_ldx\propto\delta_{k,l} \end{equation} where $\...
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Inner Product Examples, what is the points?

Example: For $ -\pi<x<\pi$, $$x =-2 \sum_{n=1}^{\infty} \frac{(-1)^n}{n} \sin(nx)$$ and $$x^3 =-2 \sum_{n=1}^{\infty} \left( \frac{\pi^2}{n}-\frac{6}{n^3} \right)(-1)^n \sin(nx)$$ by ...
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Non-strict column diagonally dominant matrix inner product

Let $A \in \mathbb{R}^{n \times n}$ be a normalized non-strict column diagonally dominant matrix, that is: $$a_{j,j} = \sum_{i \ne j} \left|a_{i,j}\right|$$ where $$0 \le a_{j,j} \le 1$$ and $$-...
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Space of $f(0)=f(1)$: Is Hilbert space?

Let $S$ be space consisting of collection of square integrable continuous functions $f:[0,1]\rightarrow\mathbb{R}$ with the constraint $f(0)=f(1)$. So $S$ is an inner product space with the inner ...
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Estimates on the integral of an inner product

Let $X$ be an inner product space. For vector-valued functions $F = (f_1,f_2), G = (g_1,g_2): [0,1] \to X^2$, we define the inner product $$(F, G) = \int_0^1 f_1g_1 + f_2g_2.$$ In particular, $$ ||F||...
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27 views

What would be a basis of $L^2(\Omega )$

Let $(\Omega ,\mathcal F,P)$ a probability space and $$L^2(\Omega )=\{random\ variable\ X\mid \mathbb E[X^2]<\infty \}$$ is a vector space. What would be a basis of $L^2(\Omega )$ ? I also know ...
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20 views

The set of Riemannian metrics on a submanifold of $\mathbb{R}^{n}$

Consider a subset $U \subset \mathbb{R}^{n}$. Clearly, $U$ can be considered as a smooth ($n$-dimensional) submanifold of $\mathbb{R}^{n}$. A Riemannian metric on $U$ is a smooth map $g$ which ...
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Hilbert space is orthornormality needed for representation?

In a Hilbert space $H$ with countable basis, if I know there is a countable basis $\{h_n\}$ of $H$ then can I express every element $h\in H$ therein as: \begin{equation} h = \sum_n \langle h,h_n\...
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Two inner products in one vector space.

Please, can you help me answer the some following questions? In theory functional analysis. At first, I want to consider finitely dimensional vector space V over field K(real or complex). Now, if it ...
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Gram-Schmidt & Inner Space Question?

I've been confused about this question from my homework and was hoping if someone could tell me how to do them, or if there are any substitutions to be made in these types of problems? (a) In the ...
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37 views

Hermitian adjoint

I'm trying to solve this task, but I'm not sure, if my solution for a) is correct. For b), i dont find a starting point. Did someone have an idea how to solve this? Thanks in advance. Be $V$ the set ...
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$X$ be real i.p.s. dim.>1 , if two closed balls,none of which is a subset of the other,intersect then do the boundaries of the balls intersect too?

Let $X$ be a real inner product space of dimension more than $1$ , let $B[x;r] , B[y;s]$ be two closed balls having non-empty intersection where none of the balls is a subset of the other , then is ...
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generalization of positive-definite matrices to matrices over finite fields

Let $\mathbb{F}$ be a field, $\mathbb{F}^n$ be the $n$-dimensional vector space over $\mathbb{F}$, and $M_{n\times n}(\mathbb{F})$ be the space of $n\times n$ matrices with entries in $\mathbb{F}$. We ...
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30 views

In a complex vector space with inner product: If $\langle T(v),v \rangle=0$ for any $v \in V$ then $T=0$

I'm having a hard time trying to prove the proposition of the title. In the accepted answer to this post written by Christopher A. Wong he mentions the following variation of the polarization identity:...
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25 views

Calculate the adjoint map

i'm trying to solve this task, but I don't find a starting point. Did someone have an idea how to solve this? Be V the set $\{f \in \mathbb{R}[X]| grad\,f \leq 2 \}$. This becomes to an euclidic ...
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scalar product for $\mathbb Q$-vector space in $\mathbb C$

In the texbooks I have for linear algebra, the scalar products are only introduce for $\mathbb R$ and $\mathbb C$ vector space, that lead me to following question: $W:=span(1,\sqrt{2}) \subset \...
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If $U$ is a vector subspace of a Hilbert space $H$, then each $x∈H$ acts on $U$ as a bounded linear function $〈x〉$. Is $x↦〈x〉$ injective?

If $H$ is a $\mathbb R$-Hilbert space, then the duality pairing $$\langle\;\cdot\;,\;\cdot\;\rangle_{H,\:H'}:H\times H'\;,\;\;\;(x,\Phi)\mapsto\Phi(x)$$ can be considered as being a mapping $H\times H\...
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Prove that there is a unique $v \in V$ for which $\langle v,v_k\rangle=c_k$ for all $k$

Let $V$ be a finite dimensional vector space over $\mathbb{R}$, with fixed basis $B=\{v_1,\dots,v_n\}$. Suppose $\langle u,v\rangle$ is an inner product on $V$. If $c_1,\dots,c_n$ are arbitrary ...
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Time (only) dependence with respect to the inner product of a wave function in $L^2(\mathbb{R})$

In my book "Quantum Theory for Mathematians" By B. Hall there is a discussion about the derivative of the inner product of a time-dependent wave functions $\psi(t)$ (note: no position dependence is ...
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Dimension and separability of $\ell^2(I)$?

Let $\ell^2(I)= \left\{ x:I\rightarrow \mathbb C\mid \sum _{i\in I} |x_i|^2<\infty \right\} $ with inner product $\sum_{i\in I}x_i \bar y_i$. I am supposed to find the dimension of this space and ...
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41 views

Prove the completion of the span of $ \left\{ e^{i\lambda t} \right\} _{\lambda\in \mathbb R}$ is not separable

Let $G$ be the span of $ \left\{ e^{i\lambda t} \right\} _{\lambda\in \mathbb R}$ with inner product $$ \left\langle f,g \right\rangle =\lim _{T\rightarrow \infty}\frac 1{2T}\int_{-T}^Tf\bar g .$$ I ...
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25 views

Does the Parseval identity imply the completeness of an orthonormal system?

Let $V$ be an inner product space which is not complete. Can there by an orthonormal system satisfying the Parseval identity for each vector but which is not complete i.e the only vector orthogonal to ...
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33 views

Find $\inf _{a,b,c\in \mathbb C}\int _{[-\pi,\pi]}|x+a+b\cos x+c\sin x | ^2dx$

I need to find $\inf _{a,b,c\in \mathbb C}\int _{[-\pi,\pi]}|x+a+b\cos x+c\sin x | ^2dx$. I thought about using the fact $\int _{[-\pi,\pi]}f\bar g$ is an inner product $ \left\langle f,g \right\...
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31 views

Show quadratic form is positive definite

If the quadratic form of $Q(x)=x^T Ax$ and $\langle x,y\rangle=\frac{1}{2}[Q(x+y)-Q(x)-Q(y)]$. Where $x, y$ are vectors in $\def\R{\Bbb R}\R^n$ and $A$ is a $n\times n$ matrix. Show that $\langle\, , \...
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Seemingly strange exercise on unitary Hermitian operator

Let $V$ be a finite dimensional inner product space. Let $T$ be a Hermitian unitary operator. Prove there's a subspace $W$ such that for each $v\in V$ we have $Tv=w-w^\prime$ where $w\in W,w^\...
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Matrix representing inner product?

Is is true that given an inner product space $ \left\langle -,- \right\rangle $, there's a matrix $A$ such that $ \left\langle u,v \right\rangle =\left\langle Au,Av \right\rangle _\mathbb{R^n}$? I ...
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How is defined the inner product $g_p$ on $T_p \mathbb{R}^n/\Gamma$ at the point $p$?

In the book Eigenvalues in Riemannian Geometry of Isaac Chavel page $28$, I have some questions related to the resolution of the spectrum of the tori. The lattice acts on $\mathbb R^n$ by $$γ(x)...
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22 views

Calculating adjoint operator for standard inner product of matrices

Let $V=M_2(\mathbb R )$ with $ \left\langle A,B \right\rangle =\operatorname{tr}(B^tA)$. Define $$ T\begin{pmatrix}a& b\\c& d\end{pmatrix}=\begin{pmatrix}3d& 2c\\-b& 4a\end{pmatrix}. ...
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Projections satisfying $\| Px-Qx \| <\|x \|$ for nonzero $x$?

Let $V$ be a f.d inner product space with subspace $M,N$ and corresponding orthogonal projections $Q,P$. I need to prove that if $\| Px-Qx \| <\|x \|$ for all nonzero $x$, then $\dim M=\dim N$. As ...
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Can $f(x,z) = x^Tx + \sum\limits_{i = 1}^n \dfrac{x_ix_i}{z_i}$ be written with multiple inner products at the same time?

I am running into a very interesting phenomenon that I do not quite understand (Illustration of an example of so called subset of $\mathbb{R}^n$) For example, suppose we have a subset of $X \...
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Prove $\Bigg(\langle\nabla f(x),x \rangle = af(x) \Bigg) \Leftrightarrow \Bigg(f(tx)=t^af(x) \Bigg)$ for $f: \Bbb R^n \to \Bbb R$ differentiable

Let $n\in \Bbb N , a \in \Bbb R$ and $f: \Bbb R^n \to \Bbb R$ differentiable. I have to show: $$\Bigg(\langle\nabla f(x),x \rangle = af(x) \:\:\:\forall x \in \Bbb R ^n \backslash\{ 0 \} \Bigg) \...
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Inner product induced norm vs $l_2$ norm

Some related problems: Relationship between inner product and norm What norm Induced inner product? My problem comes from one step of a certain proof: $\|Av\|^2=(Av)^T(Av)=v^TA^TAv=v^TIv =...
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If $H$ is a Hilbert space, $U≤H$ is closed and $E≤U^⊥$ such that $x∈H$ with $x⊥_H E$ implies $x∈U$, then $U^⊥=\overline E^{\langle\;⋅\;,\;⋅\;\rangle}$

Let $\left(H,\langle\;\cdot\;,\;\cdot\;\rangle\right)$ be a separable Hilbert space $U$ be a closed subspace of $H$ $E$ be a subspace of $$U^\perp:=\left\{x\in H:\langle x,u\rangle=0\text{ for all }...
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Inner-product on skew-hermitian matrices

Let $$\mathfrak{u}(n)=\{X\in M(n,\Bbb C):X+X^*=0\}$$ where $X^*$ is the conjugate transpose. Then, $\mathfrak{u}(n)$ is a real vector space. Problem. Show that $\langle X,Y\rangle=\...
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How can find the vector that satisfy some conditions

I have a question Assume that there are 3 vectors x1,x2,x3 (each vector has the size 3*1 (3 dimension)) I want to find these vector that satisfy below conditions (the ininitial assumption x1 = [1 0 ...
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If $\ \|v\|^{2}=\sum \langle v | v_{i} \rangle^2 $ for every $v$ then the set $\{v_{i}\}$ is orthogonal

Suppose that $\{v_{1}\cdot\cdot\cdot v_{n}\}$ are unit vectors in $\mathbb{R}^{n}$ such that $$\ \|v\|^{2}=\sum_{1}^{n} |\langle v | v_{i} \rangle|^{2}, \forall v\in\mathbb{R}^{n}.$$ Then how to ...
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Expressing the orthogonal projections on a linear operator $T$'s eigenspaces as polynomials in $T$

In the inner product space $\mathbb{C}^{2}$ with its standard inner product, let $$ T\begin{pmatrix} x\\y \end{pmatrix} = \begin{pmatrix} 3x+4y\\-4x+3y \end{pmatrix} $$ a linear operator. Express the ...
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34 views

Show that following three statements are equivalent

Proposition: Let $V$ be finite-dimensional inner-product space and $A\in L(V)$. Show that following three statements are equivalent:1) A is hermitian operator.2) For every orthonormal basis $b$ in $V$ ...
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Estimating Lorentzian inner product

Let $\mathbb{L}^{n+1}$ be the Lorentz space, that is, the Euclidean space $\mathbb{R}^{n+1}$ equipped with the nondegenerate bilinear form $$ \langle x, y\rangle = x_1 y_1 + \cdots + x_n y_n - x_{n+1}...
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I don't understand theorem about hermitian operators

Theorem: Let $V$ be finite-dimensional inner-product space and $A\in L(V) $. There exist unique operator $A^*$ such that $\langle Ax,y \rangle=\langle x,A^*y \rangle$ for every $x,y\in V$. Proof: ...
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37 views

If $Tv=\mu v$ for some $\mu>0$, then $v\in\ker(T^{1/2})^\perp$

Let $V$ be a separable $\mathbb R$-Hilbert space $T$ be a bounded, linear, nonnegative and symmetric operator on $V$ $(v_n)_{n\in\mathbb N}$ be an orthonormal basis of $V$ with $$Tv_n=\mu_nv_n\;\;\;\...
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If $ι:U→V$ is a Hilbert-Schmidt embedding and $(v_n)_{n∈ℕ}$ is an orthonormal basis of $V$, then $(ιι^*v_n)_{n∈ℕ}$ is an orthonormal basis of $ιU$

Let $(U,\langle\;\cdot\;,\;\cdot\;\rangle_U)$ and $(V,\langle\;\cdot\;,\;\cdot\;\rangle_V)$ be separable $\mathbb R$-Hilbert spaces $\iota:U\to V$ be a Hilbert-Schmidt embedding $T:=\iota\iota^\ast$ ...
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71 views

Show that $\langle x,y\rangle_A = \langle Ax,Ay\rangle$ is an inner product on $\mathbb R^n$

Let $A$ be an $n \times n$ matrix with real enteries. Define $\langle x,y\rangle_A = \langle Ax,Ay\rangle, \quad x,y \in \mathbb R^n$ , where $\langle,\rangle$ is a standard inner product on $\mathbb ...
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1answer
55 views

Proof about isometries

i'm trying to prove this statements, but I don't find a starting point. Did someone have an idea how to prove this? Thanks in advance. Be $V=R^n$ furnished with the standard inner product and the ...
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63 views

Given $Q:ℝ^d→(\text{Hilbert-Schmidt operators }U→ℝ^d)$, find a Hilbert-Schmidt operator $T:U→L^2(ℝ^d,ℝ^d)$ with $Q(x)u=(Tu)(x)$

Let$^1$ $U$ be a separable $\mathbb R$-Hilbert space $d\in\mathbb N$ $\lambda$ denote the Lebesgue measure on $\mathbb R^d$ $\Omega\subseteq\mathbb R^d$ be a bounded domain $H:=L^2(\Omega,\mathbb R^...
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1answer
29 views

existence of inner product preserving linear map?

I want to prove this: Given a vector space $V$ on $\mathbb{R}$ with a positive definite inner product $\left \langle .,. \right \rangle$. Show that there exist a natural number $p$ and a linear map $...
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56 views

Inner product on Complex Space - Proof

Studying an introduction to Hermitan inner products and complex spaces, I've found my self stuck to deal with a rather than classic example of an inner product. The complete exercise goes as follows :...
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36 views

Proof that $A=\lambda U$ [duplicate]

Let $A \in L(V)$ where $V$ is finite-dimensional inner-product space such that $\langle x,y \rangle = 0 $ implies $\langle Ax,Ay \rangle = 0 $. Show that there exist $U$ and $\lambda$ such that $A=\...