Questions tagged [inner-products]

For questions about inner products and inner product spaces, including questions about the dot product.

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In a finite dimensional inner product space with $T ∈ L(V)$, show that $\langle u,v\rangle = \langle T(u),T(v)\rangle$ implies $T$ is invertible.

Here is how I've tried to go about it, and I'm curious if it's true or if I'm way off base. T is invertible iff null$(T)=\{0\}$. Let $v∈V$ and suppose $T(v)=0$. If we can show that $v=0$, then $T$ is ...
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Prob. 10, Sec. 3.2, in Erwin Kreyszig's "Introductory functional analysis with applications"

Here is Prob. 10, Sec. 3.2, in the book Introductory Functional Analysis With Applications by Erwin Kreyszig: ... Let $T \colon X \to X$ be a bounded linear operator on a complex inner product space $...
Saaqib Mahmood's user avatar
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Axler "Linear Algebra Done Right" Exercise 6.B.13

This exercise appears in Section 6.B "Orthonormal Bases" in Linear Algebra Done Right by Sheldon Axler. Inner product spaces, norms, orthogonality, and orthonormal bases have been ...
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Inner product space of measures

Let $(X,\Sigma)$ be measurable space and $\mu_1,\mu_2,\dots$ set of finite measures on $X$ such that $\mu_i \perp \mu_j$ for $i\neq j$. Now we can consider space of measures: $$ \mathcal{M} = \left\{ ...
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Find the adjoint under the inner product $\langle f, g \rangle = \int_0^1 f(t)g(t)t \ dt$ of $\mathcal{L}(f)(t) = \frac{d^2 f}{dt^2} + f$.

Find the adjoint under the inner product $\langle f, g \rangle = \int_0^1 f(t)g(t)t \ dt$ of $\mathcal{L}(f)(t) = \dfrac{d^2 f}{dt^2} + f$ with $f(0) = 0$ and $f'(1) = 0$. Note: The weight function is ...
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Hermitian positive semi-definite matrix is a Gram matrix

I showed that every Gram matrix, i.e. a $n \times n$ matrix $A$ with $A_{ij} = \langle x_i,x_j\rangle$ where $x_1,...,x_n$ are vectors in an inner product vector space $V$, is Hermitian and positive ...
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Incomplete inner product space has a subspace such that the direct sum with orthogonal isn't the whole space

I would like to show that for any incomplete inner product space $H$ there exists a closed subspace $H_0$ such that $H_0 + H_0^{\bot}\neq H$ Here and there vere slightly relevant discussion. I saw an ...
Lada Dudnikova's user avatar
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Geometric interpretation of Isotropic vector

For real inner product space $(V,\langle.,.\rangle)$ there is a $\Bbb C$-bilinear form $( , )$ on $V\otimes\Bbb C$. This extension gives rise to a Hermitian inner product, again denoted by $( , )$ on $...
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Is it possible to define an inner product to an arbitrary field?

I've been trying to find the most general definition of an inner product space. Every definition I've found is either to $\mathbb{R}$ or to $\mathbb{C}$. Is it possible to define an inner product to ...
Syd Kerckhove's user avatar
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An inner product on the dual space of a non-complete inner product space?

As is well known, for any Hilbert space $V$, there is a natural inner product on the continuous dual. (the space of all continuous linear functionals). Is there a way to endow an inner product on ...
Asaf Shachar's user avatar
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Duality pairing and difference with inner product in Hilbert spaces

My question is an extension to the post How is the acting of $H^{-1}$ on $H^1_0$ defined?. Here duality pairings were discussed and even given explicit examples. Let $U$ and $V$ be Hilbert spaces ...
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Does $X \times \mathbb{R} \simeq X$ hold for infinite dimension inner product space $X$?

The $X$ is infinite dimension real inner product space, not restricting it as a Hilbert space. This question has troubled my friend and me a lot of days. It is obviously true when $X$ is infinite ...
CTuser_103's user avatar
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Answer clarification for why the parallelogram law implies that the norm is induced by an inner product.

In short: If a norm $\Vert\cdot\Vert$ on a real vector space satisfies the parallelogram law, and $\langle x,y\rangle := \frac14\left(\Vert x+y\Vert^2 - \Vert x-y\Vert^2\right)$, then how can we show ...
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The miraculous nature of the "matrix coefficients" $\langle Tv,v \rangle$ (especially in the context of positive type functions)

$\newcommand{\ak}[1]{\langle #1 \rangle}$I've noticed that for linear operators $T$ and an inner product $ \ak{\bullet, \bullet }$, the expression $\ak{ Tv,v}$ tends to show up a lot. For instance, it ...
D.R.'s user avatar
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Is the Euclidean norm canonical?

In the spirit of this and this question, I'm interested in the motivation for defining the Euclidean norm in $\mathbb R^n$ to be $\|x\|=\sqrt{\sum_ix_i^2}$. Of course, Euclidean geometry provides a ...
WillG's user avatar
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Is the natural norm on the exterior algebra submultiplicative?

For an inner product space $(V,\langle \cdot,\cdot\rangle)$ the exterior algebra $\Lambda V$ inherits an inner product, which satisfies $\langle a_1\wedge \dots \wedge a_n,b_1\wedge\dots\wedge b_n\...
Jan Bohr's user avatar
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Does projections to orthogonal summands closed $\implies$ subspace closed?

Let $X$ be a inner product space, with orthogonal decomposition $X=V \oplus W$. Give $X$ the topology induced by the norm induced by the inner product. Let $E\subset X$ be a subspace such that the ...
Chi Cheuk Tsang's user avatar
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Defining an inner product by means of a multilinear functional

Let $V$ be a complex vector space and an anti-linear involution $J:V \rightarrow V$ (this means that $J^2 = I$ and if $\lambda \in \mathbb{C}$ and $x, y \in V$ we have $J(\lambda x + y) = \overline{\...
Affonso's user avatar
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Dense Subspaces and Orthonormal Bases.

I haven't seen the following result stated as such, so I'm wondering if my proof is correct. I'd appreciate feedback. Let $H$ be an infinite dimensional separable Hilbert space, and $D$ a proper ...
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Inner product for dual space

Suppose we have a Hilbert space $H$. Is there any explicit expression for the inner product on $H^*$ without resorting to Riesz representation theorem? I am NOT looking for one that uses the ...
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Orthonormal basis of polynomials

I am trying to find an orthonormal basis of the vector space $P^{3}(t)$ with an inner product defined by $$\langle f, g \rangle = \int_0^1f(t)g(t)dt$$ by applying the gram schmit alogorotin to ${(...
Quality's user avatar
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Geometric interpretation of $x_1^2y_1^2+x_2^2y_2^2+x_3^2y_3^2+\dots$

Say $x$ and $y$ are two $L_2$ unit vectors of size $n$. In that case the inner product: $$x_1y_1+x_2y_2+x_3y_3+\dots+x_ny_n$$ Is the cosine of the angle between them. For an application I was ...
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Homework: Second derivative of $\langle Ax, x \rangle$

So let $A \in M_{n}$ and define $f: \mathbb{R}^n \to \mathbb{R}$ by $f(x) = \langle Ax, x \rangle $. Find f' and f''. After some work, I found the first derivative to be $f'(x)(v) = \langle Ax, v \...
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Prove or disprove this argument

Let $L>0$ and let $\Omega$ be the set of all integrable functions from $[0,L]$ to $]0,+\infty[$. For all $\varphi, \psi \in \Omega$ define $\left \langle \varphi,\psi \right \rangle:=\int_{0}^{L}\...
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Inner Product as Weighted Average

Let $\zeta=e^{2\pi i/n}$, where $n\geq3$. Let $||\cdot||$ be the norm induced by the complex inner product $\langle\cdot,\cdot\rangle$. Then $$\langle x,y\rangle=\frac{1}{n}\sum_{k=1}^{n}||x+\zeta^...
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Exercise $5$, Section $6.A$ - Linear Algebra Done Right

Exercise: Suppose $V$ is finite dimensional and $T\in L(V)$ is such that $\|Tv\|\le \|v\|$ for every $v\in V$. Prove that $T-\sqrt{2}I$ is invertible. Proof: We will prove the contrapositive. Suppose ...
Seeker's user avatar
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Optimization with inner product condition

I designed an optimization problem to improve the performance of the neural network learning process on the gradients. I spent hours, but unfortunately it did not work out. Given vectors ${\bf a}_1, {\...
jack negs's user avatar
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For $v_1,\dots ,v_n\in\mathbb{C}^n$ define $A=(\langle v_i,v_j\rangle)_{i,j=1}^n$. Prove that $A$ is a non negative linear operator on $\mathbb{C}^n$.

Consider $\mathbb C^n$ as an inner product space with the standard inner product $\langle \cdot, \cdot \rangle$. For $v_1,\dots ,v_n\in \mathbb{C}^n$ define the $n\times n$ matrix $$A=(\langle v_i,v_j ...
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Minimize sum of maximum dot product between unit vectors

Let $\mathbb{S}_{d-1}$ denote the unit $(d-1)$-sphere. Let \begin{align} f(d, n) = \operatorname*{argmin}_{x \in (\mathbb{S}_{d-1})^n} \sum_{i \in n} \max_{j \in n - \{i\}} x_i \cdot x_j \end{...
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If $x$ maximizes $x^T(u+v)$ subject to $\|x\|^*\le 1$, then $x^T u \ge x^T v \Leftrightarrow \|u\| \ge\|v\|$

Suppose that $\|\cdot\|$ is a norm on $\mathbb{R}^n$ and $\|\cdot\|^*$ is its dual norm. Consider two vectors $u$ and $v$ in $\mathbb{R}^n$ such that $u+v\neq 0$. Let $x$ be a vector that maximizes $x^...
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Geometric Motivation for Inner Product

I think some background will make the kind of answer I'm looking for clearer. I'm trying to think of an elementary proof of the Pythagorean Theorem. I don't like the geometric proofs because they all ...
Charles Hudgins's user avatar
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Physical meaning of the dot product of a vector and its laplacian

What is the physical meaning of $$\boldsymbol{A}\cdot (\nabla^2\boldsymbol{A})$$ where $\boldsymbol{A}$ is a vector field in 3D space? What does it show?
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Maclaurin series as an orthonormal expansion of a Hilbert space

Let $\mathbb{L}$ be the set of all entire functions, so every $f \in \mathbb{L}$ can be rewritten as a Maclaurin series: $$f(x)=\sum_{n\geq0} \frac{f^{(n}(0)}{n!}x^n.$$ I'm tring to come out with a ...
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On the monotonicity of angles between dual basis vectors

I want to show that if I make the angles between basis vectors $e_1,...,e_n\in\Bbb R^n$ smaller, then the angles between the dual basis vectors $e_1^*,...,e_n^*\in\Bbb R^n$ become larger. More ...
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In what spaces are outer product of x with itself positive semidefinite?

In the case of vectors in $\mathbb R^n$, it is quite simple to see that for any vector $x$, $$ v^Txx^Tv = (v^Tx)^2 \geq 0$$ so clearly the form $xx^T$ must form a positive semidefinite matrix. But ...
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A construction of the Hodge Dual operator

This question about showing that an alternative construction of the Hodge dual operator satisfies to the universal property through which the Hodge dual is usually defined. Let me give the ...
Jules Pitcho's user avatar
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How to expand the right-hand side of $\langle x,y \rangle = \frac{1}{N} \sum_{n=0}^{N-1}\lVert x+\alpha^n y \rVert^2\alpha^n$?

If $\langle \cdot, \cdot \rangle$ is a complex inner product and if $\alpha \in \mathbb{C}$ and $\alpha^N = 1$ but $\alpha^2 \not = 1$, then show that $$\langle x,y \rangle = \frac{1}{N} \sum_{n=0}^{...
user547265's user avatar
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Divergence of Petersson inner product

Consider a complex number $z = x + i y$ and functions $f,g : \mathbb{H} \to \mathbb{C}$ (where $\mathbb{H}$ is the upper half-plane i.e. complex numbers whose imaginary part is greater equal to zero). ...
Gorbz's user avatar
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Equivalence of hermitian forms under subgroups of $\textrm{GL}_n(\mathbb{C})$

Let $X \in \textrm{GL}_n(\mathbb{C})$ be a hermitian matrix ($\space ^t \overline{X} = X$). For another hermitian matrix $Y$, let's say that $X \sim Y$ if there exists a $g \in \textrm{GL}_n(\mathbb{...
D_S's user avatar
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"Projecting" one convex set onto another (though actually, it's just a translation).

In linear algebra, we learn that we can project a vector $x \in \mathbb{R}^n$ onto a linear subspace $A \subseteq \mathbb{R}^n$. I have hunch that this can be generalized considerably. In particular, ...
goblin GONE's user avatar
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Strong convexity and the Legendre transform

Suppose that I have a strongly convex function $f(\mathbf{x}): \mathbb{R}^m \rightarrow \mathbb{R}$. Is the Legendre transform of this function also strongly convex? As far as I can tell, strict ...
ReverseMathematics's user avatar
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Progressed : Convergence problem in Hilbert Space and necessity of inner product

******** PROGRESS : so thanks to Ian's great comment I can get by the proof and that completeness is necessary but I need to know does this hold for general Banach spaces that are not Hilbert spaces? ...
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Is the inverse of a bijective connectedness preserving map , on a complete real inner product space , also connectedness preserving?

Let $X$ be a complete real inner-product space and $f:X \to X$ be a bijection which maps connected sets to connected sets ; then is it necessarily true that $f^{-1}$ also maps connected sets to ...
user avatar
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If for every operator represented by $A$ w.r.t to a basis $\mathcal{B}$, the matrix representation of $T^*$ is $A^*$, then $\mathcal{B}$ is orthogonal

Let $V$ be a finite-dimensional inner product space. Assume that for every linear operator $T$, represented by $A$ w.r.t to a basis $\mathcal{B}$, the matrix representation of the adjoint w.r.t to $\...
JonTrav's user avatar
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Application of Cauchy-Schwarz with Sobolev norms

I'm working through the problems in the initial value formulation chapter in Wald's General Relativity. A short summary of the problem. I have to show that $$\sup_{x\in A}|f(x)|\le C||f||_{A,k}$$ ...
Ryan Unger's user avatar
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4 votes
2 answers
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Inner product alternative definition

I'm trying to make an alternative (but equivalent) definition of an inner product. I prefer to use arbitrary sums with $\sum_i v_i$ instead of sum of just two vectors $v+w$, and few but clear axioms. ...
dami's user avatar
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Orthonormal basis Parsevals identity.

Let $O={u_1,...,u_k}$ be an orthonormal set in $V$. Prove that $O$ is an orthonormal basis if and only if Parseval's identity holds for all $v,w \in V$ i.e if and only if $$\langle v,w\rangle=\sum_{...
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Using inner product property to determine if operator is an isomorphism.

Let $\varphi$ be an operator on a $k$-vector space $V$ with an inner product $\langle\cdot,\cdot\rangle$. Suppose that $\langle v,\varphi v\rangle = 0$ for every $v\in V$. If we take $k=\mathbb R$, is ...
user avatar
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Existence of $n$ distinct (real) roots of an orthogonal polynomial

I'm trying to get my head around the proof that an orthogonal polynomial ($P_n$ say) has at least n distinct roots. My understanding of the proof http://en.wikipedia.org/wiki/Orthogonal_polynomials#...
Tim Green's user avatar
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Why is the order reversed in the conjugate symmetry axiom for the inner-product?

The conjugate symmetry axiom for the inner product is as follows: $$\overline{\langle x,y\rangle } = \langle y,x\rangle$$ It is my understanding that the conjugate is there so that the norm of a ...
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