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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Self adjoint differentiation operator on vector space of polynomials and inner products.

Let $n \in \mathbb{N}$. For the following, consider the vector space of polynomials of degree less than or equal to $n$ with complex coefficients, $\mathcal{P}_n(\mathbb{C})$. a) If we equip the ...
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“Adding” inner products

Is there a general rule to simplify things like $<x,y> - <x,z>$ or generally $<.,.> \pm <.,.>$ I cant find anything in my notes that talks about this.
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Vector inequality $(\langle Ax,y \rangle^2) \le (\langle Ax,x\rangle * \langle Ay,y\rangle)$

Let $A = (a_{ij})_{1\le i \le j \le n} \in M_{n \times n}^{\Bbb R}$ such that $$(a_{ij})=\begin{cases} 2, i = j \\ 1, i\ne j \end{cases}$$ Prove that $(\langle Ax,y \rangle^2) \le (\langle Ax,x\...
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Finding an orthonormal basis for a matrix

If we have a subspace W of $\mathbb{R}^2$ spanned by $(3,4)$. Using the standard inner product, let E be the orthogonal projection of $\mathbb{R}^2$ onto W. Find an orthonormal basis in which E is ...
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53 views

Hermitian Pairings from Positive Functionals

Let $A$ be $*$-algebra and $\phi:A \to {\mathbb C}$ a positive linear functional, that is, one for which $\phi(aa^*) \geq 0$, for all $a \in A$. When does it hold that a symmetric sesquilinear form, i....
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Inner product property proof

Let $V$ be a vector space. Prove that if $\forall\beta \in V,\langle\alpha,\beta\rangle=0$ then $\alpha=0$. I couldn't think of any way to prove this directly, so I tried the contrapositive. Which ...
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Is the adjoint of an isometry $S$ the inverse of $S$?

Let $S$ be an isometry with adjoint $S^\ast$. Prove that $S^\ast S=SS^\ast=I$. Here is what I have so far: Since $S$ is isometry, there is an orthonormal basis of $e_j$'s such that $\|Se_j\|=\|e_j\|...
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On the proof of $\langle T(v),v \rangle = 0$ for all $v \in V \iff T(v)=0$ for all $v \in V $

For a complex inner product space $V$ and a linear map $T:V \to V$ $$\langle T(v),v \rangle = 0 \text{ for all } v \in V \iff T(v)=0 \text{ for all } v \in V $$ One proof makes use of the identity $...
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33 views

Find the extrema of $\sum_{i=1}^n u_i v_i \log \left| \frac{v_i}{u_i} \right|$

This question is similar to the following one: Maximizing and minimizing dot products. However there are significant differences, hence I opened a new question. Maximize and minimize $$\sum_{i=1}^n ...
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60 views

Maximizing and minimizing dot products

Given 2 vectors $u,v \in \mathbb{R^n}$ such that $\|u\| = 1$ and $\sum_{i=1}^n v_i= c$ where $c<1$, I would like to maximize $$\sum_{i=1}^n u_i v_i \log (v_i)$$ and minimize $$\sum_{i=1}^n u_i v_i \...
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50 views

If $\{u, v\}$ is an orthonormal set, how is $\|u - v\| = \sqrt{2}$?

Let $V$ be a vector space and for $u,v\in{}V$ let $\langle{}u,v\rangle$ define an inner product on $V$. If {$u,v$} is an orthonormal set in $V$, then $\|u-v\|=\sqrt{2}$ Somehow I'm getting $\sqrt{-2}$...
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1answer
38 views

Solution to Systems of Equations, Orthogonal Polynomials

Let $\mu_j=\int_a^b x^jw(x)dx$ be the $j$th moment of the weight distribution $w(x)$. Show that the linear system of equations $$\left[\begin{matrix} \mu_0 & \mu_1 & \cdots & \mu_{n-1}...
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1answer
46 views

Non singular matrix $M$ such that $MAM^T=F$, normal form

Show that there is a non-singular matrix $M$ such that $MAM^T = F$ for any antisymmetric matrix $A$ where the normal form $F$ is a matrix with $2 \times 2$ blocks on its principal diagonal which are ...
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1answer
35 views

Let ($\pi, V$) be a representation of $G$ with character $X$: if $\langle X, X\rangle=2$ then $V$ is the sum of two irreducible representations

Let $(\pi, V)$ be a representation of $G$ with character $X$. Prove that if $\langle X, X\rangle=2$ then $V$ is the sum of two irreducible representations I was under the impression that the inner ...
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1answer
33 views

Continuity of inner product and change of limit order

First of all, please note that the specific context of ergodic theory could possibly not matter and this could reduce to a simply a question about Hilbert Space. As a part of a proof i'm working on, ...
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Is an Infinite Dimensional Inner Product Space Isomorphic to its Continuous Dual Space?

What I know so far is In finite dimensions the dual space is continuous, has the same dimension as the space, and the "dual basis" is in fact a basis. the Riesz representation theorem proves the ...
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Invertible matrix and orthogonal matrix proofs

So the question asks: $V$ is a $2$-dimensional inner product space with bases $B = \langle b_1,b_2\rangle$ and $C =\langle c_1,c_2\rangle$. Let $A_B$ be the $2\times 2$ matrix whose $(i, j)$-entry is $...
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$\int_{A}f^2 = 0$ implies $f(x) = 0 \text{ } \forall x \in A$

Follow-up question to Cauchy-Schwarz for integrals. Here's the basic idea: over the vector space of real-valued square-integrable functions defined on a subset $A \subset \mathbb{R}$, define the ...
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Cauchy-Schwarz for integrals

One of the exercises (3.2) of Izenman's Modern Multivariate Statistical Techniques is for $A \subset \mathbb{R}$, $$\left( \int_{A}fg\right)^2 \leq \left(\int_{A}f^2\right) \left(\int_{A}g^2\right)$$ ...
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Finding if the orthonormal basis of P1 with a given inner product

Hi, So I know for a basis to be orthonormal, the inner product needs to be 0 between components. So the standard basis <1,t> would not work in this case, as the integral from 1 to zero of t is t^...
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$V^⊥$ where V is spanned by ${cos x, cos^2 x, cos^3 x, cos^4 x, . . . }$

So this is the last question of the problem, it asks: Let $C^0$ be the vector space of continuous functions on the interval $[−2, 2]$. Consider this an inner product space with the inner product $<...
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$\langle u,v\rangle = [u]^T_B A_B[v]_B$ definition

My textbooks says: An inner product $\langle u,v\rangle = [u]^T_B A_B[v]_B$ with bases $B = \langle b_1, b_2\rangle$ So does it work for all $u,v ∈ V$? I don't see any proof or further explanation ...
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point evaluation cannot be inner product of continuous functions

Suppose we consider the vector space of real-valued continuous functions $V:= C[-1,1]$. Then $\phi(f):= f(0) \; \forall \; f \in V$ is a linear functional. I want to show that $\phi(f)$ cannot be ...
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39 views

Inner Product Notation

I'm using a mathematics research paper to write code that calculates the minimum distance between two polygons. While I understand this isn't a code site, my questions is focused on how to read ...
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1answer
24 views

Determining when a linear transformation is an inner product

Hi, I know that $$ \langle v,w\rangle $$ satisfies the first three conditions of being an inner product always. As such, the only condition that needs to be met is $$ T(v) \cdot T(v) > 0 $$ ...
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Functions satisfying 4 out of 5 inner product properties

Let us consider function $s:K^m \times K^m \mapsto K$ (here $K = \mathbb{R}$ or $K = \mathbb{C}$). If $\forall x, y, z \in K, \forall \lambda \in K$ $s(x + y, z) = s(x, z) + s(y, z)$ $s(\lambda x, ...
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How do I show that if is normal, then if T is a projection then it must be an orthogonal projection?

T is an operator on a finite dimensional inner product space. How do I show that if T is a projection, then it must be an orthogonal projection? I know I have to use the fact that $T^*T=TT^*$, and I ...
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orthonormal basis in $L^2$ space

Let $\{\phi_i (x)\}_{i=1}^\infty$ be an orthonormal basis for $L^2 (S)$. Prove that $\{\psi_{ij} (x,y) = \phi_i (x) \phi_j (y)\}_{i,j=1}^\infty $ is an orthonormal basis for $L^2 (S \times S)$. Thanks ...
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Can a functional be expressed by an inner product in an infinite-dimensional space?

Question first, motivation afterwards. Let $P$ be the vector space of all real polynomials (with no restriction on the degree). Is there an inner product $\langle p\!\mid\!q\rangle$ on $P$, and a ...
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A list is a basis if norm of difference with orthonormal basis is bounded by given constant less than unity

This is a problem from Linear Algebra Done Right, 3rd Edition, problem 14 in 6.B. Given $\{e_1, \dotsc, e_n\}$ is an orthonormal basis of $V$, and $||v_j -e_j|| < \frac{1}{\sqrt{n}} \; \forall \; ...
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Projection of function onto subspace spanned by non orthogonal bases

Can this be done using the inner product? I can successfully do this by applying the Gram-Schmidt process and orthonormalizing the bases, but what if I wanted to project the function onto my non-...
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Generating Orthogonal polynomials for Gaussian Quadrature

I am attempting to show that if $u_j=\int_a^b w(x)x^jdx$ and $$A_n=\left( \begin{array}{cccc} u_0 & u_1 & ... & u_n \\ u_1 & u_2 & ... & u_{n+1} \\ \vdots & \vdots & \...
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Proving that $tr(A^*A)=tr(B^*B)$

$A, B \in M_n(F)$ are unitarily equivalent. How do I use that to prove that $tr(A^*A)=tr(B^*B)$? Additionally, how would I use that fact to prove that $\sum_{i,j}|A_{ij}|^2=\sum_{i,j}|B_{ij}|^2$?
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Projection of inner product spanned by orthonormal basis

So the question asks: Consider the inner product $⟨f,g⟩=$$ \begin{align} \int_{-1}^{1} f(x)g(x) \ dx &\end{align}$$ $ on $P_2$, the space of all polynomials of degree 2 or less. Find the ...
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Vectors and dot product in spherical coordinate system [duplicate]

Let $\vec{v_1}$ and $\vec{v_2}$ given: $\overrightarrow{V_1} = r_1\hat{u_r} + \theta_1\hat{u_\theta} + \phi_1\hat{u_\phi} \\ \overrightarrow{V_2} = r_2\hat{u_r} + \theta_2\hat{u_\theta} + \phi_2\hat{...
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Property of the conjugate transpose matrix with inner product

I'm trying to prove that for a certain matrix $A$, and its conjugate transpose $A^*$, we have $⟨Ax,y⟩=⟨x,A^*y⟩$, where $⟨⟩$ represent the inner product. So here it's simply the dot product in $R^n$. ...
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the complement subspace of a subspace of an ips

$\langle p,q\rangle=\int_{-1}^1p(x)q(x)\,\mathrm{d}x \quad, (p,q\in R_3[x])$\,where $R_3[x]$ is the vector space of all real polynomial of degree less than equal to 3. If $W=\{\,p \in R_3[x]:p(0)=0\}$...
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Why does the Cauchy-Schwarz Inequality even have a name?

When I came across the Cauchy-Schwarz inequality the other day, I found it really weird that this was its own thing, and it had lines upon lines of proof. I've always thought the geometric definition ...
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A conjugate LT in an inner product space

Let $V$ be a finite dimensional complex inner product space. Let $J$ be a conjugate linear map from $V$ to $V$ such that $J^2 =1$. Can we say $\langle Ju, Jv \rangle = \langle v, u \rangle$ for all ...
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V complex IPS and LT with real matrix entries

V is finite dimensional complex inner product space. There exist a basis such that matrix entries of T in wrt this basis are real. Can we say there will exist an orthonormal basis wrt which matrix ...
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How do I prove that $\left \langle T(x),x \right \rangle=0,\forall x\in V\Rightarrow T=T_0$?

T is a linear operator on V, with an adjoint. V is a complex inner product space, not necessarily of finite dimension. How do I prove $$\left \langle T(x),x \right \rangle=0,\forall x\in V\...
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How can there be an inner product space when inner product yields a scalar? [closed]

I thought the inner product yields a scalar in both real and complex cases. How can a space be made up of scalars? Taking two vectors $a$ and $b$, $a=(a_1,a_2)$ and $b=(b_1,b_2)$, the inner product is ...
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Prove that $R(T^{*})^\perp =N(T)$

Let V be an inner product space, with T being a linear operator on V. How do I prove that $R(T^{*})^\perp =N(T)$? I tried setting $x\in R(T^{*})$ and $Ty\in N(T)$, and set up an inner product = 0 ...
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Let $V$ be an inner product space. Prove $\langle x, 0\rangle = \langle 0, x\rangle = 0$ for $x \in V$. [closed]

Let $V$ be an inner product space. For $x \in V$, prove $\langle x, 0\rangle = \langle 0, x\rangle = 0$.
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Show that $\langle y_i, y_j\rangle = 0 \forall i \neq j.$

Let $y_1, y_2, . . .$ be a sequence in a Hilbert space. Let $V_n$ be the linear span of $\{y_1, y_2, . . . , y_n\}$. Assume that for $n \ge 1, ∥y_{n+1}∥ \le ∥y − y_{n+1}∥$ for each $y \in V_n$. ...
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Finding the orthogonal projection of a given vector on the given subspace W of the inner product space V.

Let $V = R^3$ with the standard inner product $u = (2,1,3)$ and $W = \{(x,y,z) : x + 3y - 2z = 0\}$ I came up with the basis $\{(-3,1,0), (2,0,1)\}$ but these are not orthogonal to each other. I'm ...
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1answer
41 views

Separation of a convex set and a vector by a hyperplane

How a hyperplane separates a convex set. Please give me mathematical and geometrical representation and also I want to know the consequences after separations? Actually, I was reading a scholarly ...
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If $\langle x,y\rangle = \langle x,z\rangle$ , then $y=z$ PROOF

I'm trying to prove a property of inner products. The property is: $$\langle x,y\rangle=\langle x,z\rangle \forall x \in V\Rightarrow y=z$$ My proof: Let $y=z+w$ then $$\langle x,z+w\rangle = \...
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Is my proof correct (normal operator)?

Q. Let V be a $n-$dimensional space over $\mathbb{C}$ and $T$ a normal operator. Prove that all T-invariant space is $T^{*}$-invariant. A. If $T$ is normal over $\mathbb{C}$ then $V$ admit a basis of ...
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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 ...