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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Intersection of nested closed bounded convex sets in Euclidean space

I read that in a complete Euclidean space - i.e. a normed real space with the norm induced by the scalar product - any sequence of nested bounded non-empty closed convex sets has a non-empty ...
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43 views

Showing the conjugate symmetric property of an inner product when we don't know if our field is $\mathbb{C}$.

The conjugate symmetric property of an inner product states that $\langle{x, y}\rangle = \overline{\langle{y, x}\rangle}$. My question is regarding showing this when we don't necessarily know that our ...
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1answer
30 views

Let $A$ be an invertible , prove that there exist positive constants $c_1$ and $c_2$ such that $c_1X^tX\leq X^tA^tAX\leq c_2X^tX $

Let $A$ be an invertible $n \times n$ matrix over $\mathbb{R}$, prove that there exist positive constants $c_1$ and $c_2$ such that $$c_1X^tX\leq X^tA^tAX\leq c_2X^tX $$ for all $X \in ...
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63 views

Does $S^\bot+T^\bot = (S\cap T)^\bot$ hold in infinite-dimensional spaces?

If $S$ and $T$ are subspaces of some finite-dimensional inner product space then $$S^\bot+T^\bot = (S\cap T)^\bot.$$ See, for example, this post or this post Does it hold in infinite-dimensional ...
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What is the difference between an isometric operator and a unitary operator on a Hilbert space?

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 the identity operator) What is the difference between ...
4
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2answers
79 views

Let $W_1$ and $W_2$ be subspaces of a finite dimensional inner product space space. Prove that $(W_1 \cap W_2)^\perp=W_1^\perp + W_2^\perp $

Let $W_1$ and $W_2$ be subspaces of a finite dimensional inner product space space. Prove that $$(W_1 \cap W_2)^\perp=W_1^\perp + W_2^\perp $$ My Try One direction is easy : Let $\alpha \neq 0$ ...
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1answer
28 views

Understanding a bilinear form problem from Greub's Multilinear Algebra

I read the following problem from exercise sets of Greub's Multilinear Algebra, Chapter I, Sec. 1 Let $E$, $E^*$ be a pair of dual spaces and assume that $\mathit{\Phi}:E^{*}\times E\to\Gamma$ is ...
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Show that $(Au,Bv)=(u,A^tBv)$

Let $ A, B $ be matrices of order $ n $, and $ \vec{u}, \vec{v} $ vectors from euclidean space $ \mathbb{R}^n $, then $ (Au,Bv) = (u,A^tBv) $ pd. $(\cdot ,\cdot)$ is my notation for inner product, ...
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92 views

Question about existence of unitary matrices with certain properties

We are given a set of $d$ normalized vectors on a $d$-dimensional complex vector space: $e_1$, $e_2$... $e_d$, where $$\langle e_j,e_j\rangle=1$$ for all $j$. These are not necessarily mutually ...
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84 views

How can I prove that the span of an a subspace and it's orthogonal complement is the whole vector space?

The book Linear and Geometric Algebra explains the following theorem in a way that I haven't been able to figure out: If $\mathbf{A}$ and $\mathbf{B}$ are subspaces of a vector space $\mathbf{B}$ ...
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1answer
78 views

Find the orthogonal projection of $f(x)=4x^2−4$ onto the subspace spanned by $g(x)=x−12$ and $h(x)=1$.

Use the inner product $\langle f,g\rangle =\int_0^1 f(x)g(x)dx$ in the vector space $C^0[0,1]$ to find the orthogonal projection of $f(x)=4x^2−4$ onto the subspace $V$ spanned by $g(x)=x−1/2$ and ...
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1answer
172 views

Show that the sup-norm is not derived from an inner product

I am trying to show that the norm $$\lVert{\cdot} \rVert _{\infty}=\sup_{t \in R}|x(t)|$$ does not come from an inner product (the norm is defined on all bounded and continuous real valued ...
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2answers
78 views

How to find orthonormal basis for inner product space?

In $\mathbb{R}^3$ we declare an inner product as follows: $\langle v,u \rangle \:=\:v^t\begin{pmatrix}1 & 0 & 0 \\0 & 2 & 0 \\0 & 0 & 3\end{pmatrix}u$ How can I find an ...
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58 views

question about inner product and $f^*$

In $\mathbb{R}$3 we declare an inner product as follows: $\langle v,u \rangle \:=\:v^t\begin{pmatrix}1 & 0 & 0 \\0 & 2 & 0 \\0 & 0 & 3\end{pmatrix}u$ we have operator $f ...
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75 views

Fourier coefficients with respect to an orthonormal basis for an inner product space

$V = \operatorname{span}(S)$, where $S = \{(1, i, 0), (1 - i, 2, 4i)\}$, and $x = (3 + i, 4i, -4)$. Apply the Gram–Schmidt process to the given subset $S$ of the inner product space $V$ ...
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2answers
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If $\|Tv\|=\|T^*v\|$ for all $v\in V$, then $T$ is a normal operator

I have solved a question but I am not sure the last step of the question. If someone can verify it that would be great. Let $V$ be a finite dimensional vector space with complex inner product. Let ...
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1answer
119 views

Project sin(x) onto orthonormal basis that span ${(1, x, x^2, x^3, x^4, x^5)}$ on domain $[-\pi, \pi]$

I am self-studying LA through Linear Algebra Done Right 2nd ed. I probably made a blatant error somewhere but I have been stuck for a whole day now. The book gave the answer $0.987862x − 0.155271x^3 ...
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1answer
38 views

How do the inner products on $L^2$ look like?

I was wondering whether all scalar products in $(L^2[0,1],\lambda)$ are given by $\langle f,g \rangle := \int f(x)g(x) \cdot w(x) d\lambda(x)$? If this is true, what are the exact conditions that we ...
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47 views

Defining an inner abstract vector space

Since an inner product space is an abstract vector space with an additional structure called an inner product, and this additional structure is a component wise operation that associates each pair of ...
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1answer
55 views

The inner product determines the structure of the space

The Hilbert space employs inner product to determine the geometric structure,e.x. the angle. But I couldn't understand how. For example, the key structure of Euclidean space $\mathbb{R}^2$ is that it ...
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251 views

Prove the inequality $\sum_{i,j = 1}^n {{A_{i,j}}({x_i}^2 - {x_i}{x_j})} \ge 0 $

A is a square matrix with positive elements and x is a real vector (both of them n>1 dimensional). Prove that for any such matrix and vector $$\sum\limits_{i,j = 1}^n {{A_{i,j}}({x_i}^2 - {x_i}{x_j})} ...
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1answer
40 views

weighted inner product of polynomials, can weight function be complex?

I am just learning about inner-products on polynomial space, where the coefficients of the polynomials may be complex: $P_m(\mathbf{F})$ The inner-product given by: $\langle p,q \rangle = \int_0^1 ...
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1answer
41 views

A question on linearity of inner product

The linearity of inner product on $(X,\langle.,.\rangle)$ is usually written as $$\langle x+\alpha y,z\rangle = \langle x,z\rangle + \alpha\langle y,z\rangle,\qquad \forall (\alpha,x,y,z)\in R\times ...
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2answers
67 views

Is a norm on $R^n$ linear?

I was reading the book Linear Algebra Done Right by Axler. In the chapter on inner product space (Ch.6), he defines the norm of x on $R^n$ space as: $||x|| = \sqrt{x_1^2 + ... + x_n^2}$ and says: ...
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1answer
61 views

Why $\langle a,x\rangle = \langle b,x\rangle,\forall x\in X\implies a=b$ [closed]

Let $X$ be (possibly infinite-dimensional) Hilbert space. How can we show that if $$\langle a,x\rangle = \langle b,x\rangle,\forall x\in X$$ then $a=b$?
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66 views

Rigid motion on $\mathbb{R}^2$ which fixes the origin is linear

Let $V=\mathbb{R}^2$ be an inner product space with the standard inner product, and let $T$ be a rigid motion of $V$. Suppose $T(0)=0$, prove that $T$ is linear. (A rigid motion of an inner product ...
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Area preserving transformation in a higher dimensional space is unitary.

In $\mathbb{R}^3$, a linear operator $Q:\mathbb{R}^3 \to \mathbb{R}^3$ preserves the area of parallelograms: that is, given $x,y\in \mathbb{R}^3$, the area of a parallelogram formed by $x$ and $y$ is ...
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2answers
66 views

Show that $\langle x, Ax \rangle + \langle b, x \rangle = c$ can be transformed to $\langle x', Ax' \rangle = 1$

Let $A$ be a real, regular, symmetric $n \times n$ matrix, $b \in \mathbb{R}^n$ and $c \in \mathbb{R}$ How can I show that $$\langle x, Ax \rangle + \langle b, x \rangle = c$$ can be transformed by ...
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36 views

Express Norm Using Inner Product

I'd like to know whether there's a way to express a norm using inner product, for example , is there any inner product we may use that is equal to $(||Ax-b||_2)^2$ ? Thanks in advance.
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0answers
20 views

Bessel-like inequality

Let $\{e_n\}$ be an orthonormal sequence in an inner product space E. Then I'm trying to show the following inequality: $$\sum_1^\infty| \langle x, e_n \rangle \langle y, e_n \rangle | \leq ...
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35 views

Transformation of a sum of dot products

I'm not quite sure about this, So I'd like if someone could help me. Can someone explain to me how they get from 2 to 3? \begin{align}\left<2u,u+v\right> &= 0\tag1\\ ...
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1answer
21 views

Inner product with the given property and vector distance problem

Is there an inner product in the $P_2$ ( polynomials, $deg(p)\leq 2$)inner product space so that $\{1,t,t^2\}$ is an orthonormal base? Also, how do you find a subspace for a given vector and a ...
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1answer
25 views

Why do we have $\Vert y_n\Vert^2=\langle x;y_n\rangle$

I can't see why the following equalities are valid: Let $X$ be an inner product space with an othonormal system $e_n, n\in\mathbb{N}$ and $x\in X$. Define $y_n=\sum_{i=1}^n\langle ...
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1answer
39 views

Few basic things unclear to me about inner product spaces and orthonormal basis

Few things unclear to me about inner product spaces: assume V is an inner product space with B orthonormal basis. Why is it true that: $$\langle x,y\rangle = \langle[x]_{B} , [y]_B \rangle{st}$$ ...
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1answer
74 views

Why is the Cauchy-Schwarz inequality considered to be so important?

I've read in the book "Linear Algebra done right" by Axler that the Cauchy-Schwarz inequality is one of the most important results in mathematics. However, in what the book covers and what we have ...
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2answers
43 views

Given a symmetric matrix A, find an orthogonal matrix S such that $S^TAS$ is a diagonal matrix

Given the symmetric matrix: $$A = \left( \begin{array}{ccc} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 1 \\ \end{array} \right)$$ find an orthogonal matrix $S$ such that $S^TAS$ is a ...
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1answer
43 views

Help with integral (inner product of stochastic and deterministic process)

i need to calculate an integral of the form $$ X = \int_0^T w(t) \sin (\omega t) dt $$ where $w(t)$ is a stochastic normal process (white noise), $\sin(\omega t)$ is deterministic. How do I do that? ...
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1answer
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Give an example of a spanning set of $\ell^2(N)$ which is also a Bessel sequence but not a frame for $\ell^2(N)$

We know that in a finite dimensional Hilbert space, every spanning set is a frame, but this is not true for infinite dimensional space. It is easy to find an example which is a spanning set but not a ...
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1answer
33 views

Find the signature of $Q(x_1,\ldots,x_n)= \sum_{i,j=1}^{n} a_ia_jx_ix_j$

In $\mathbb{R}^n$ let $Q(x_1,\ldots,x_n)= \sum_{i,j=1}^{n} a_ia_jx_ix_j$ quadratic form. $a:=(a_1,\ldots,a_n)\neq0$ $\in \mathbb{R}^n$ find the signature of $Q$
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1answer
44 views

Linear algebra: determining if something is an inner product space

If I have a potential inner product space over P2, where $\left< p, q\right> = p(0)q(0)$ How do I determine whether or not it is an inner product space? Using the four axioms I have: ...
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1answer
82 views

Determinant (or positive definiteness) of a Hankel matrix

I need to prove that the Hankel matrix given by $a_{ij}=\frac{1}{i+j}$ is positive definite. It turns out that it is a special case of the Cauchy matrices, and the determinant is given by the Cauchy ...
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2answers
28 views

Proving the image of inner product map is whole subspace

I'm doing a specimen exam question and they often have typos and missed pieces of necessary information. I think the question I'm doing might be one such example, but am not sure: We're given that ...
4
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1answer
42 views

Is a function $f$ with $f(X)\perp (I-f)(X)$ necessarily linear?

Let $X$ be a real or complex inner-product space, and let $f : X\rightarrow X$ be a function such that every element of $f(X)$ is orthogonal to every element of $(I-f)(X)$. Prove or give a ...
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Show that $V = U^\perp \bigoplus U$

If $(V,\langle , \rangle)$ is a Euclidean vector space, $U \subseteq V$ is a subspace of V and $U^\perp := \{v \in V | \langle v,u \rangle = 0, \forall u \in U\}$. Show $V = U^\perp \bigoplus U$ In ...
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1answer
65 views

Orthogonal projection in Inner product space

Let V be $n$-Dimensional ($n\ge1$) inner product space . Let $T:V \rightarrow V$ be a linear map which maintains $ T^2=T$ , $\forall v \in V\ ||Tv||\le||v||$. Prove that there is exists a subspace ...
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1answer
84 views

Symmetric matrix over inner product space

I try really hard to prove this Question. let $A_{nXn}(\mathbb{R})$ Symmetric matrix $A=A^t$ let $\lambda$ be the greatest Eigenvalue of A. we will define over the field $\mathbb{R}$ with the ...
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1answer
47 views

Sufficient condition for two operators being identical on Hilbert space

Considering two bounded linear operators $S,T$ in $\mathcal{B}(X)$, where $X$ is a complex Hilbert space. If $\def\norm#1#2{\langle {#1},{#2}\rangle} \norm{Sx}{x} = \norm{Tx}{x}$ for all $x\in X$, do ...
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3answers
65 views

Consider a set $S$ of unit vectors in $\mathbb R^2$ such that $\left<x,y\right>=-\frac12$ if $x,y\in S,x\ne y$.

This is a question from an entrance exam paper. Consider a set $S$ of unit vectors in $\mathbb R^2$ such that $\left<x,y\right>=-\frac12$ if $x,y\in S,x\ne y$.Then it is necessarily true that ...
0
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1answer
170 views

Apollonius’ Identity inner product space

$||z-x||^2+||z-y||^2=\frac{1}{2}||x-y||^2+2||z-\frac{x+y}{2}||^2$ I proved it by expanding both sides and i found both sides are equal. Are there any easy way to prove it?
4
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1answer
96 views

An inequality for inner product space: $\|x-z\|.\|y-t\|\leq \|x-y\|.\|z-t\|+\|y-z\|.\|x-t\|$

In a inner product space show that the following inequality holds. $\|x-z\|.\|y-t\|\leq \|x-y\|.\|z-t\|+\|y-z\|.\|x-t\|$ I am stuck in proving this inequality