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Questions tagged [inner-product-space]

An inner product space is a vector space equipped with an inner product. The inner product is a generalization of the “dot” product often used in vector calculus.

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if $ A \in R^{n \times n}$ , $A > 0$ and $ b \in R^n$ then the function $\frac{1}{2}\langle Ax,x\rangle - \langle b,x\rangle$ is convex in $R^n$

Show by direct estimates that if $ A \in R^{n \times n}$ , $A > 0$ and $ b \in R^n$ then the function $$\frac{1}{2}\langle Ax,x\rangle - \langle b,x\rangle$$ with $x$ is convex on $R^n$. My ...
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Is the norm of a vector given by dividing it by the square root of the inner product with itself?

I've known the norm (aka. unit vector in the direction of) a vector (let's call it $v$) to be given by the formula: $v/\sqrt{v \cdot v}$, where $\cdot$ is the dot product. But can this be ...
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Representing a transformation's matrix as an inner product

Let $V$ be an inner product space, $T: V \to V$ a linear map, and $A=M(T,P,P)$ for an orthonormal basis $P=[v_1,...,v_n]$ (where $M(T,P,Q)$ is the matrix representation of $T$ with respect to $P$ and $...
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Differential Equation by Minimization

Suppose we want to solve $u + xu' = 0$, which has the general solution $u = \frac{C}{x}$, by minimizing the length squared of $u + xu'$. This should work due to the positive definite condition of ...
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For $\langle Tx, x \rangle \geq \|x\|^2$, prove a solution exists for $Tx = y$.

Edit I've posted this a couple other times, so I now plan on deleting those, and just using this one. Here's the original problem: $H$ is a real Hilbert space. Let $T: H\longrightarrow H$ be a ...
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Are there trilinear inner products?

Is there such a thing as a "trilinear inner product"? The definition of an inner product is: Let $H$ be a vector space over $\mathbb{K}\in \{\mathbb{R,C}\}$. An inner product is a map $\langle \...
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Inner Product of two functions

The inner product of two vectors $\vec a$ and $\vec b$ of n dimensions, is given by, $$(\vec a \,, \vec{b}) =a_1b_1 + a_2b_2 + a_3b_3 + \,\,...\,\,+a_nb_n $$ If a function is considered to be a ...
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Which one of the following option satisfies (1) and (2)?

Consider the subspaces $W_1$ and $W_2$ of $\mathbb R^3$ given by $W_1=\{(x,y,z)\in \mathbb R^3: x+y+z=0\}$ and $W_2=\{(x,y,z)\in \mathbb R^3:x-y+z=0\}$. If $W$ is a subspace $\mathbb R^3$ such that $...
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Alternate inner products on Euclidean space?

After reading about inner products as a generalization of the dot product, I was hoping to be able to prove that the dot product is in some sense the unique inner product in Euclidean space (e.g., up ...
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Calculating inner product using Parsevals identity

Letting $x$ and $y$ be elements of some separable Hilbert space $\mathcal{H}$, we know that for any orthonormal basis $\{e_i\}_{i=1}^\infty$, we can write $$ x = \sum_{i=1}^\infty \langle x, e_i \...
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Convergence to zero in $L^2$ by this sequence of continuous functions

I have been struggling with the following exercise: Let $\{f\}_n\subseteq C[a,b]$ be a cauchy sequence w.r.t the $L^2$ norm, s.t. for every $[c,d]\subseteq [a.b]$ we have $\lim_{n\to \infty} \int_c^d ...
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How to find a counter-example for the 4th Axiom of Scalar Products?

I was given the following $\langle U, V \rangle = 3U_1V_1 + 2U_1V_2 + 2U_2V_1 + U_2V_2$ and I was asked to demonstrate if its a scalar product or not. I've tested the first three axioms and they ...
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Hermitian vs Frobenius inner products?

(Correct me if I am wrong but) I have concluded that the Frobenius inner product (ip) is the generalization of the complex vectors ip to that of matrices. My q is wrt whether or not the Hermitian ip ...
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Express vector $\vec{w}$ in terms of dot product of vectors from basis $(\vec{v_k})$ of $n$ dimensional Euclidean space

For given basis $(\vec{v_k})$ of Euclidean space $\mathbb{E}^n$ find representation of vector $\vec{w}$ expressed by dot product $<\vec{v_k}, \vec{w}>$. In other words coefficients of $\vec{w}$ ...
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Orthogonal vector to a plane using a different inner product

Considering $\mathbb{R}$ with the inner product $$\langle(a_1,a_2,a_3),(b_1,b_2,b_3)\rangle=2(a_1b_1+a_2b_2+a_3b_3)-(a_1b_2+a_2b_1+a_2b_3+a_3b_2)$$ Then, how could we find the set of vectors ...
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How to find the rotation angle and axis of rotation of linear transformation?

I need some help with this problem: We know that $T(x_1,x_2,x_3)=(x_2,x_3,x_1)$. We suspect that it is a rotation matrix, to be sure, we need to determine the rotation axis and the rotation angle. ...
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Difficulties with Inner products and polarization identities

I am discussing the general inner product space. Here is what Polarization Identities mean. I denote the inner product by $(x,y)$. I am having a difficult time with the polarization identities. Of ...
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How to check this statements about the inner product?

I need help with this problem: Determine which of the following statements are true or false, justify your answer. If $x$ and $y$ are vectors on $\mathbb{R}^n$, then whe have: $x\cdot y=0$ ...
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Prove norm on any Hilbert space is strictly convex

I've seen in lecture notes that norm on any Hilbert space is strictly convex means "$\|x\|=\|y\|=1, \quad\|x+y\|=2 \Rightarrow x=y$" But why this means strict convexity? I thought strict convexity ...
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Question about Inner product in polynomials

Let $V=\{p(x); \partial(p(x)<2\} \cup\{0\}$ a vector space, determine a inner product such that the basis $\{1,x,\frac{x^2}{2!}\}$ is orthonormal. My solution: I found the inner product: $<...
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Dual pairing and inner product in a Hilbert space (and in $L^2(V)$)

I put beforehand that there are some similar questions in this blog, but I nonetheless would like to pose my question as I did not find any explanatory answer. Let us consider a vector space H, ...
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Axiomatize inner product spaces in terms of angles?

Inner product space is a vector space $V$ over a field $F$ together with an inner product $P:V\times V\to F$ that satisfies the inner product axioms. This inner product induces an angle $\angle (v,w) ...
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Variants of The Polarization Identity

A problem in Steele's Cauchy Schwarz Master Class asks the reader to prove these "variants of the polarization identity". Let $\langle \cdot, \cdot \rangle$ be a complex inner product and $\alpha \in ...
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Finding orthonormal basis of subspace spanned by two functions

From S.L Linear Algebra: Let $V$ be the subspace of functions generated by the two functions $f$, $g$ such that $f(t)=t$ and $g(t)=t^2$. Find an orthonormal basis for $V$. In this case, $V$ is ...
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some properties of inner product space

Let $V$ be an inner product space and $u,v,w\in V$. Choose the correct option $|\langle u,v\rangle| \le || u|| + ||v||$ $|\langle u,v\rangle| \le \frac{1}{2}(|| u||^2 + ||v||^2)$ $|\langle u,v\...
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Dual norm of the norm induced by inner product

Given a positive definite matrix A, let $<x, y>_A=x^\top Ay$. This inner product induces a norm $\|x\|_A^2=<x, x>_A = x^\top A x$. My question is, what is the dual norm of $\|\cdot\|_A$? ...
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Inner Product Differentiation Rule

The dot product differentiation rule is $(\vec f(t) \cdot \vec g(t))' = \vec f\ '(t) \cdot \vec g(t) + \vec f(t) \cdot \vec g\ '(t)$, which simplifies to $$(\vec f(t) \cdot \vec f(t))' = 2(\vec f\ '(t)...
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Find a unique vector $y$ such that $g(x)=<x,y> $ for all $x \in V$ (Riesz Representation Theorem example)

Consider $g : M (R)_{2x2}$ → $R$ given by $g(A)=a_{11} + 2a_{12} + 3a_{32} +4a_{22}$. We consider on $M_{2x2} (R)$ the inner product given by $<A,B> = tr(A^t ,B)$. Find the vector $y$. I only ...
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Prove the max value of a linear functional is the length of its Riesz vector

$\mathcal{V}$ is an inner product space over $\mathbb{F}$ of finite-dimension, and $\phi : \mathcal{V} \to \mathbb{F}$ is a linear functional. Let $\textbf{r}$ be the Riesz vector for $\phi$. Prove ...
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Is this a valid proof for the Cauchy-Schwarz inequality?

QUESTION: Show that equality holds in the Cauchy-Schwarz inequality |⟨x, y⟩| ≤ ∥x∥ ∥y∥ . for x, y if and only if x and y are linearly dependent. I have written up this but I not sure if it is a full ...
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Given 2 different orthogonal bases, prove that projection does not depend on the choice of basis.

That is (working over $\mathbb{R}^n$), if we have 2 different orthogonal bases, $\mathcal{B} = \{b_1, ...b_n\}$ and $\mathcal{C} = \{c_1, ...c_n\}$, such that $$T_1(v) = \frac{\langle v, b_1\rangle}{\...
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Calculate $\inf _{a, b, c \in \mathbb{R}} \int_{-1}^{1}\left|x^{3}-a x^{2}-b x-c\right|^{2} d x$

Calculate $$\inf\limits_{a, b, c \in \mathbb{R}} \,\int_{-1}^{1}\left|x^{3}-a x^{2}-b x-c\right|^{2} \mathrm d x$$ I am new to Hilbert space, I see similar questions used the formula: $\langle f, g\...
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Symmetry of inner product

Peter J. Cameron's "Notes on Linear Algebra" defines An inner product on a real vector space $V$ is a function $b: V \times V \to R$ satisfying b is bilinear (that is, b is linear in the first ...
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Show that equality holds in the Cauchy-Schwarz inequality |⟨x, y⟩| ≤ ∥x∥ ∥y∥ . for x, y if and only if x and y are linearly dependent…

Could someone please help me with this proof? I have written up this but I not sure if it is a full proof, since it is an if and only if statement. Could someone read and inform me where I can go ...
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Show that equality holds in the Cauchy-Schwarz inequality |⟨x, y⟩| ≤ ∥x∥ ∥y∥ . for x, y if and only if x and y are linearly dependent.

So, I have written up a proof but I not sure if it is a full proof, since it is an if and only if statement. Could someone read and inform me where I can go from here if there is more to prove? So, ...
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Does the orthogonal complement determine the inner product up to scaling?

Let $V$ be a real $n$-dimensional vector space, and let $g,h$ be two inner products on $V$. Fix some $1\le k\le n-1$, and denote by $\text{Gr}_k(V)$ the Grassmannian of $k$-dimensional subspaces of $V$...
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Spectral norm and inner product

We know that for a general $N\times n$ matrix $A$, its spectral norm is defined as $$\|A\|=\sup_{x\in S^{n-1}} \|Ax\|_{2},$$ where $x$ is from the unit sphere. My question is that why when $A$ ...
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How is the trace of the adjoint of Lie algebra elements defined? [duplicate]

Consider an element $X\in\mathfrak g$ of some Lie algebra $\mathfrak g$. I understand that $\mathfrak g$ can be represented via its action on other elements of the same algebra, as $\operatorname{ad}(...
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Do the $|$ around $|\langle u,v\rangle|$ refer to absolute value in the inner product version of the Cauchy-Schwarz inequality?

The full inequality is: $|\langle u,v\rangle| \leq ||u|| ||v||$ I understand that $||$ around the vectors $u$ and $v$ signifies the taking of their norm, but what do the single | around $\langle ...
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Projection of $\mathbb{R}^2$ onto the line $y=x$, but NOT with respect to the usual inner product.

$$T(x,y) = (2x-y, 2x-y)$$ is a projection of $\mathbb{R}^2$ onto the line $y = x$, but not with respect to the usual inner product. Is it an orthogonal projection with respect to some inner product? ...
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Inner Product Question from 6.1 of Friedberg Insel Spence

Above is the problem given. I couldn't solve it so looked at the solution and this was the first step: If I take this step for granted, I can solve the problem, but my question is: how did they get ...
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Integrating Factors Via Minimization

I am trying to algorithmize an idea to find integrating factors for inexact differential equations via optimization. Currently, I'm sticking to simple cases that can already be solved by other means. ...
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Prove $\langle u,v \rangle = \frac{1}{4} \sum^4_{k=1}i^k \left \Vert u+i^kv \right \Vert$

For an assignment in one of my math classes I have this problem. Here is where I have gotten so far. This is in V, an inner product space over $\mathbb{C}$ $$\frac{1}{4} \sum^4_{k=1}i^k \left \Vert u+...
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Why does $\mathscr{P}_2$ being isomorphic to $R^3$ imply that a constantwise addition of objects in $\mathscr{P}_2$ is an inner product?

Why does saying that $\mathscr{P}_2$ is isomorphic to $R^3$ immediately prove that a constant-wise addition of objects in $\mathscr{P}_2$ is an inner product? What I mean by constant-wise addition ...
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Scalar Products and Projections

Proposition: Let w $\in V$ so that V is a vector space over $\mathbb{R}$ and $||w|| \neq 0$. For every v in V there exists a unique $c\in \mathbb{R}$ so that $v-cw$ is perpendicular to w. My Proof: ...
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lower bound for determinant of $(X^TY)$

Am looking for a lower bound for determinant, $\det(X^TY)$ where $X^T$ is $p \times n$ and $Y$ is $n \times p$. Is it $Tr(X^TY)^{-1}$? Regardless, what are other lower bounds for this? $X,Y$ are real-...
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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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Does closed convex sets having unique nearest points imply the parallelogram law?

It's a well-known result that if $X$ is a Hilbert space, then for any closed convex subset $C$ of $X$, there exists a unique element of $C$ with minimal norm. I'm wondering whether the converse is ...
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Kernel function with a feature space equipped with an inner product that is not the dot product

Premise: A function $K: \mathbb R^d \times \mathbb R^d \to \mathbb R$ is called a kernel function on $\mathbb{R}^d$ if there exists a Hilbert space $\mathcal{H}$ and a map $\phi: \mathbb R^d \...
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Proof no nonnegative function satisfies 3 integrals (3 $L^2$ norms derived from the inner product on $C[0,1]$)

$V$ is the real inner product space $C_\mathbb{R}[0,1]$ and $a \in [0,1]$. Prove there is no nonnegative function $f \in V$ such that $$ \int_0^1 f(x) \, dx = 1,$$ $$ \int_0^1 xf(x) \, dx = a, $$ $$ \...