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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How can find the vector that satisfy some conditions

I have a question Assume that there are 3 vectors x1,x2,x3 (each vector has the size 3*1 (3 dimension)) I want to find these vector that satisfy below conditions (the ininitial assumption x1 = [1 0 ...
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36 views

If $\ \|v\|^{2}=\sum \langle v | v_{i} \rangle^2 $ for every $v$ then the set $\{v_{i}\}$ is orthogonal

Suppose that $\{v_{1}\cdot\cdot\cdot v_{n}\}$ are unit vectors in $\mathbb{R}^{n}$ such that $$\ \|v\|^{2}=\sum_{1}^{n} |\langle v | v_{i} \rangle|^{2}, \forall v\in\mathbb{R}^{n}.$$ Then how to ...
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Expressing the orthogonal projections on a linear operator $T$'s eigenspaces as polynomials in $T$

In the inner product space $\mathbb{C}^{2}$ with its standard inner product, let $$ T\begin{pmatrix} x\\y \end{pmatrix} = \begin{pmatrix} 3x+4y\\-4x+3y \end{pmatrix} $$ a linear operator. Express the ...
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Inner product (real or complex), sequence of real numbers

Suppose that $\{v_1, v_2, \dots, v_n\}$ is a basis for a vector space $V$ with inner product $\langle\cdot, \cdot\rangle$ (real or complex). Prove that for each sequence of $n$ real numbers $r_1, r_2, ...
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30 views

Show that following three statements are equivalent

Proposition: Let $V$ be finite-dimensional inner-product space and $A\in L(V)$. Show that following three statements are equivalent:1) A is hermitian operator.2) For every orthonormal basis $b$ in $V$ ...
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Estimating Lorentzian inner product

Let $\mathbb{L}^{n+1}$ be the Lorentz space, that is, the Euclidean space $\mathbb{R}^{n+1}$ equipped with the nondegenerate bilinear form $$ \langle x, y\rangle = x_1 y_1 + \cdots + x_n y_n - x_{n+1}...
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54 views

I don't understand theorem about hermitian operators

Theorem: Let $V$ be finite-dimensional inner-product space and $A\in L(V) $. There exist unique operator $A^*$ such that $\langle Ax,y \rangle=\langle x,A^*y \rangle$ for every $x,y\in V$. Proof: ...
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35 views

If $Tv=\mu v$ for some $\mu>0$, then $v\in\ker(T^{1/2})^\perp$

Let $V$ be a separable $\mathbb R$-Hilbert space $T$ be a bounded, linear, nonnegative and symmetric operator on $V$ $(v_n)_{n\in\mathbb N}$ be an orthonormal basis of $V$ with $$Tv_n=\mu_nv_n\;\;\;\...
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If $ι:U→V$ is a Hilbert-Schmidt embedding and $(v_n)_{n∈ℕ}$ is an orthonormal basis of $V$, then $(ιι^*v_n)_{n∈ℕ}$ is an orthonormal basis of $ιU$

Let $(U,\langle\;\cdot\;,\;\cdot\;\rangle_U)$ and $(V,\langle\;\cdot\;,\;\cdot\;\rangle_V)$ be separable $\mathbb R$-Hilbert spaces $\iota:U\to V$ be a Hilbert-Schmidt embedding $T:=\iota\iota^\ast$ ...
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Show that $\langle x,y\rangle_A = \langle Ax,Ay\rangle$ is an inner product on $\mathbb R^n$

Let $A$ be an $n \times n$ matrix with real enteries. Define $\langle x,y\rangle_A = \langle Ax,Ay\rangle, \quad x,y \in \mathbb R^n$ , where $\langle,\rangle$ is a standard inner product on $\mathbb ...
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1answer
52 views

Proof about isometries

i'm trying to prove this statements, but I don't find a starting point. Did someone have an idea how to prove this? Thanks in advance. Be $V=R^n$ furnished with the standard inner product and the ...
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Given $Q:ℝ^d→(\text{Hilbert-Schmidt operators }U→ℝ^d)$, find a Hilbert-Schmidt operator $T:U→L^2(ℝ^d,ℝ^d)$ with $Q(x)u=(Tu)(x)$

Let$^1$ $U$ be a separable $\mathbb R$-Hilbert space $d\in\mathbb N$ $\lambda$ denote the Lebesgue measure on $\mathbb R^d$ $\Omega\subseteq\mathbb R^d$ be a bounded domain $H:=L^2(\Omega,\mathbb R^...
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1answer
28 views

existence of inner product preserving linear map?

I want to prove this: Given a vector space $V$ on $\mathbb{R}$ with a positive definite inner product $\left \langle .,. \right \rangle$. Show that there exist a natural number $p$ and a linear map $...
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Proving that a product norm does not come from a scalar product

We have nonzero normed spaces $X_1$, $X_2$. On the product $X_1 \times X_2$ we put the norm: $$ ||(x_1,x_2)||=||x_1||_1+||x_2||_2$$ and would like to show that it does not come from an inner product. ...
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55 views

Inner product on Complex Space - Proof

Studying an introduction to Hermitan inner products and complex spaces, I've found my self stuck to deal with a rather than classic example of an inner product. The complete exercise goes as follows :...
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36 views

Proof that $A=\lambda U$ [duplicate]

Let $A \in L(V)$ where $V$ is finite-dimensional inner-product space such that $\langle x,y \rangle = 0 $ implies $\langle Ax,Ay \rangle = 0 $. Show that there exist $U$ and $\lambda$ such that $A=\...
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49 views

Existence of operator $B^2=A$

Let $A$ be hermitian operator on finite-dimensional inner-space $V$ whose eigenvalues are positive real numbers. Show that there exists hermitian operator $B$ such that $B^2=A$. Is $B$ unique?
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Inverse of hermitian operator

If $A$ is hermitian operator on finite-dimensional inner-product vector space $V$, than prove $A^{-1}$ is also hermitian operator. ( Hermitian operator $A$ is operator such that $A=A^{*}$ )
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For positive self adjoint $T$, show $|\langle Tx,y\rangle|^2 \le \langle Tx,x\rangle \langle Ty,y\rangle$

As in title, $T$ is a positive self adjoint, bounded linear operator on a Hilbert Space $X$ and I'd like to show $$|\langle Tx,y\rangle|^2 \le \langle Tx,x\rangle \langle Ty,y\rangle$$ Self adjoint ...
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Sobolev Space with partial inner product

In my work, I encountered the following problem. Consider the set of real-valued functions, which are ``balanced'', that is the set of bounded functions $f(x)$ such that $\lim_{x\rightarrow \pm \...
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42 views

Prove $\left\langle v | u \right\rangle = v\cdot (A^tAu) $ is an inner product

Let $v,u\in\mathbb{R}^n$ and $A\in M_n\left(\mathbb{R}\right)$ an invertible matrix. Prove $\left\langle v | u \right\rangle = v\cdot (A^tAu) $ is an inner product over the reals. I was able to (...
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Representation of functional on overlapping areas

I have given a functional $l$ on $C_c^\infty(\mathbb{R}^n)$. Now let's assume that for any $p \in \mathbb{R}^n$ we have a neighborhood $V_p$ and a $2\pi$-periodic $C^\infty$-function $u_p$ on $\mathbb{...
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dom(A) is a Banach space w.r.t. the Graph-norm

Let $X$ and $Y$ be Banach spaces and let $A:dom(A)\to Y$ be a linear operator, defined on a linear subspace $dom(A)\subset X $. Prof that the graph of $A$ is a closed subspace of $X\times Y$ if and ...
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32 views

When would the inner product be non symmetric over the reals?

For example the inner product where $u=(u_1,u_2) , v=(v_1,v_2)$ is $\left<u,v\right>= 3u_1v_2 + u_2v_1$ How would you prove that it is not symmetric. Thanks
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Relation between linear independence and inner product

I was given the following question: Let $V$ be an inner product space and let $u,v\in V$ be two nonzero vectors. Prove or disprove: If $\langle u,v\rangle=0$, then $u,v$ are linearly ...
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16 views

Existence of non obtuse angle of n+2 vectors in n-dimensional euclidean space.

There are n+2 distinct vectors $v_1,v_2,v_3,\cdots ,v_{n+2}$ in n-dimensional euclidean space. Prove that there must be a integer pair of $(i,j)$ which satisfies $1\leq i<j\leq n+2$, and $dot(v_i,...
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If $Z$ is a closed subset of Hilbert Space $X$, is it true that $Z\neq X \implies Z^{\perp}\neq \{0\}$?

It is clear from Projection theorem that if $Z$ is a subspace, then since $X=Z\oplus Z^{\perp}$, $Z^{\perp}$ is not trivial (By the way, is there any reasoning that would show this without referring ...
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32 views

Inner product on polynomials with complex argument

The polynomials can be given an inner product $$\int_X p(x)q^*(x)dx$$ Where X is an interval on the real number line. Consider instead, that X is a curve over the complex field with parametrization $...
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How can we compute the square root of an operator of the form $Cv=\sum_{n\in\mathbb N}\langle v,e_n\rangle_Ve_n$?

Let $\mathbb K\in\left\{\mathbb C,\mathbb R\right\}$ $U$ and $V$ be $\mathbb K$-Hilbert spaces such that $U\subseteq V$ and that the inclusion $\iota$ is Hilbert-Schmidt $C:=\iota^\ast$ denote the ...
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Hermitian product from inner product

Let $V_{\mathbb{R}}$ be a $\mathbb{R}$-vector space of dimension $n$ and let $E:=\{e_1,\dots,e_n\}$ be a base. Consider the scalar product $(\cdot,\cdot):V_{\mathbb{R}}\times V_{\mathbb{R}}\rightarrow ...
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How can we compute the adjoint of the inclusion between two Hilbert spaces?

Let $\mathbb K\in\left\{\mathbb C,\mathbb R\right\}$ $(U,\langle\;\cdot\;,\;\cdot\;\rangle_U)$ and $(V,\langle\;\cdot\;,\;\cdot\;\rangle_V)$ be $\mathbb K$-Hilbert spaces such that $U\subseteq V$ ...
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inner product and bilinear mappings

I understand that the inner product of two vectors and its properties. However I do no quite understand bilinear mappings. What is the relationship between inner products and bilinear mapping? and ...
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1answer
19 views

Equalities on a hermitian space

Let $\phi$ be a hermitian form on the hermitian space $V$ with orthonormal basis $B= \{e_1, \dots, e_n\}$. Let $T$ be an endomorphism such that for all $v \in V$, $$ Tv = \sum_{i=1}^n\phi(v,e_i)e_i. $$...
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Generalization of inner product

I was wondering if there was a widely accepted generalization of inner product spaces where the inner product look something like $\langle\bullet , \bullet\rangle:V\times V \to \mathbb{F}$, where $\...
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1answer
30 views

Riesz Representation Thereom for Polynomials with real coefficients problem

Find a polynomial q(x) $\in$ P$_2$($\Bbb R$) Such that $ p ( 1/4 ) = $$\int_0^1 p(x)q(x) \,dx$$ $. I'm sorry to ask this question, but I've been working on it for some time. The inner product on P$_2$(...
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Square-root of $\iota\iota^\ast$, where $\iota$ is an isometric embedding between Hilbert spaces

Let $U$ and $H$ be Hilbert spaces and $\iota$ be an embedding of $U$ into $H$. Then, $$\pi x:=u\;\;\;\text{for }x\in H\text{ with }x=\iota u+y\text{ for some }u\in U\text{ and }y\in\left(\iota U\right)...
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An inner product space question

Let $V$ be an inner product space and $u_1,\dots,u_k$ vectors of $V$. Let $G$ be the Gram matrix $$G_{ij}= (u_i| u_j)$$ I need to prove that if $u_1,.,u_k$ is a base of $V$ then product: $$ y^t*G*\...
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Showing $\|x+y\|^2+\|x-y\|^2=2\|x\|^2+2\|y\|^2 \Rightarrow \| \cdot \| $ is induced by scalar product

I need to show the above $\forall x,y,v \in V$ , a normed vector space on $\Bbb R$. A hint was given that i should first show that $$s:V \times V \to \Bbb R ; \: \:\: s(u,v):=\frac1 4 (\|u+v\|^2-\|u-v\...
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Zero transformation and inner product

Theorem. A linear operator $T$ on an inner product space is $0$ if and only if $\langle a,T(a)\rangle =0$ for all $a$. (Ref: Hassani, Sadri. Mathematical physics: a modern introduction to its ...
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Given a special Hilbert space $U_0$, is there a proper superspace $V$ such that the inclusion $\iota:U_0\to V$ is Hilbert-Schmidt?

Let $U$ be a Hilbert space $Q\in\mathfrak L(U)$ be nonnegative and symmetric $U_0:=Q^{1/2}U$ be equipped with $$\langle u,v\rangle_{U_0}:=\langle Q^{-1/2}u,Q^{-1/2}v\rangle_U\;\;\;\text{for }u,v\in ...
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Innerproduct of <a+b, c> = <c, c> where c != 0 [closed]

Say we have 3 non zero vectors a,b,c. If (a + b, c) = (c, c), can we conclude that a + b = c? Here is my attempt to prove the claim: a + b != 0, since (a + b, c) is non zero. I tried subtracting (c ...
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Can we find a concrete representation of $\iota\iota^\ast y$, if $\iota$ is a Hilbert-Schmidt embedding between Hilbert spaces?

Let $U$ and $H$ be real Hilbert spaces $\iota:U\to H$ be a Hilbert-Schmidt embedding $Q:=\iota\iota^\ast$ Can we find a concrete representation of $Qy$ for some $y\in H$? By Riesz' ...
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Inner products on $C_c^\infty$ corresponding to differential forms

Let $M$ be a compact, oriented Riemannian manifold of dimension $n$. We can locally identify smooth $k$-forms by smooth functions $\mathbb{R}^n \to \mathbb{C}^{m}$, where $m = \binom{n}{k}$ (via a ...
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29 views

Find an orthogonal basis for $\mathbb P_2$

The problem: For polynomials $\mathbb{P_2}$ we define the inner product between p and q as: $$ \langle p,q\rangle =p(t_0)q(t_0)+p(t_1)q(t_1)+p(t_2)q(t_2) $$ with $$t_0=0, t_1=1, \textrm{ and } ...
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1answer
17 views

Where $F$ is a complete subspace of an inner product space $X$ and $x\in X$, show there is a unique closest element of $F$ to $x$.

That is: where $F$ is a complete subspace of an inner product space $X$ and $x\in X$, prove that $\exists! z\in F$ such that $\lVert x-z\rVert = \inf_{y\in F}\lVert x-y \rVert$. Also, show that $x-z\...
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25 views

Existence of an inner product under which all elements of a finite subgroup of $GL(V)$ are isometries

I'm currently hopelessly stuck on an exercise in linear algebra and I could use some hints. Let $V$ be a finite-dimensional vector space over the reals and let $G$ be a finite subgroup of $$\...
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3answers
37 views

Explication on how obtaining $\int \langle \nabla w, \nabla w\rangle = \lambda \int \langle w, w\rangle$

Could anyone is able to explain to me how to obtain $\int \langle \nabla w, \nabla w\rangle = \lambda \int \langle w, w\rangle$ related to user7530's comment in the question : Rayleigh quotient $Q=(\...
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1answer
22 views

The signature of an inner product space does not depend on its basis

In R.W.R. Darling's "Differential Forms and Connections" an inner product is defined for a vector space $V$ as a bilinear, symmetric and nondegenerate (but not necessarily positive-definite) map from $...
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1answer
38 views

Self-adjoint operators and eigenvalues

In a previous exam the following question was asked which I was unable to answer due to the lack of knowledge of self-adjoint operators. Let $S$ be a self-adjoint operator on a real finite ...