# 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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### 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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### 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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### 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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### difference between dot product and inner product

I was wondering if a dot product is technically a term used when discussing the product of $2$ vectors is equal to $0$. And would anyone agree that an inner product is a term used when discussing the ...
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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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### 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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### Finding an inner product given a norm

Let $V=\mathbb{R}^n$. Given a norm $|\cdot |$, find the inner product for which $\langle v,v\rangle=|v|$. So basically I need to find an inner product such that $\forall v\in V. \langle v,v\rangle=1$,...
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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 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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### 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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### 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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### Prove that the eigenvalues of a real symmetric matrix are real.

I am having a difficult time with the following question. Any help will be much appreciated. Let $A$ be an $n×n$ real matrix such that $A^T = A$. We call such matrices “symmetric.” Prove that the ...
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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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### 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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### 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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### 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 ...
It seems that both isometric and unitary operators on a Hilbert space have the following property: $U^*U = I$ ($U$ is an operator and $I$ is an identity operator, $^*$ is a binary operation.) What ...