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Questions tagged [sesquilinear-forms]

A sesquilinear form is a modification of a [tag:bilinear-form], where it allows one of its arguments to be "twisted" in a [tag:semilinear] manner. One of its applications is when taking the inner product of two complex vectors. It guarantees the the result is a real number by twisting the linearity of one of the arguments by taking its complex conjugate and then performing a standard inner product with the other argument.

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Sesquilinear generalization of symplectic form

I have recently been trying to learn some basic symplectic geometry, and I have come across two sesquilinear forms which are closely related to the symplectic form. Fix $\mathbb K$ to be a field, and $...
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Polarization identity sesquilinear form

I don't understand why sometimes the polarization identity for sesquilinear form $m$ is written as $$ m(x,y)= \frac{1}{4}\sum_{n=0}^{3}i^{n}Q(x+i^{n}y) $$ with $Q(x)=m(x,x)$ and other times as $$ m(x,...
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on a complex vector space: $\|x\| = \|y\| \Leftrightarrow x+y$ orthogonal to $x-y$

I want to know if $\|x\| = \|y\|$ $\Leftrightarrow$ $x+y$ orthogonal to $x-y$. I was able to prove it for $V$ being a real vector space. However, I'm not able to work it out for complex vector spaces. ...
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sesquilinearform only small on diagonal

We are given a sesquilinear form $a \colon V \times V \to \mathbb{C}$ on a complex Hilbert space $V$. We know that $a$ is bounded and $a(u,u)$ is small for all $u \in V$. Can we make any assertions on ...
Sebastian Bechtel's user avatar
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Continuity of sesquilinear form on space of complex polynomials

I need help with the following: Prove that sesquilinear form $\Phi(p,q) = \int_0^1 p(x)\overline{q(x)}dx$ defined on space of complex polynomials on $[0,1]$, with norm $\|p\| = \int_0^1 |p(x)|dx$, is ...
Maria's user avatar
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Strongly continuous one-parameter unitary group and the form domain of the generator

Suppose $A :D(A) \subset \mathcal{H} \to \mathcal{H}$ is self-adjoint over the separable Hilbert space $\mathcal{H}$. By Stone's Theorem, $A$ generates a unique one-parameter strongly continuous group ...
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Derivative of Hermitian sesquilinear form with respect to its own matrix

Let $H$ be an $n \times n$ Hermitian matrix (in my work, it's also positive semidefinite, if that makes a difference) and $a,b \in \mathbb{C}^n$, with $\lambda(H) = \langle a \vert H \vert b \rangle$. ...
elutionary's user avatar
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Connections on indefinite inner product space bundles

Let $K$ denote a finite-dimensional vector space over $\mathbb{C}$. If $K$ is paired with a map $(\cdot,\cdot):K\times K\to\mathbb{C}$ such that, for all $\varphi,\psi,\chi\in K$ and $z,w\in\mathbb{C}$...
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Question on non-degenerate sesquilinear forms

Let $f$ be a non-degenerate sesquilinear form on $E$, is it true that for all linear $\varphi$ in $E$, so $\varphi \in E^*$, there exist $z \in E$ such that $$\varphi(x) = f(x,z)$$ and if so, why? I ...
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What it means if a functional is bounded?

I'm learning the definition of quadratic and sesquilinear forms. I found two formulations of what it means if a sesquilinear form $\tilde q$ is bounded. $|\tilde q(\phi,\psi)|\leq c||\phi||\ ||\psi||$...
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Is the polarization of quadratic form bounded?

Suppose that $q: V\times V \to \mathbb{C}$ is a bounded quadratic form, define $$ \begin{equation} \tilde q(\phi,\psi) = \frac{1}{4} [q(\phi + \psi) -q(\phi - \psi) + iq(\phi + i\psi) - iq(\phi - i\...
IGY's user avatar
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Showing the polarization of (complex) quadratic form is sesquilinear. [duplicate]

Let $q: V\times V \to \mathbb{C}$ on a complex vector space $V$ be a quadratic form. Define $\tilde q$ by the polarization identity: $$ \begin{equation} \tilde q(\phi,\psi) = \frac{1}{4} [q(\phi + \...
IGY's user avatar
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All invariant forms of (a representation of) a semi-simple lie group

Take for example, $SU(2)$. There are two well known invariant forms in the fundamental representation, namely (by the Frobenius-Schur indicator) a skew-symmetric bi-linear form: $$ \varepsilon = \left(...
Craig's user avatar
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Bound sesquilinear form $C*$-algebras

I am currently taking lectures on $C^*$-algebras. I have a question related to this topic, precisely on linear functional on these spaces. Suppose $A$ is a $C^*$-algebra, with a faithful normal state $...
Marianof's user avatar
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Are "standard" and "Hermitian" quadratic forms sub-cases of the same object?

Let $V$ be a vector space over a field $k$. The definition of quadratic form on nLab is a map $q:V\to k$ such that $q(tv)=t^2q(v)$ for all $t\in k,v\in V$, and such that the map $(v,w)\mapsto q(v+w)-q(...
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Proving operator is self-adjoint w.r.t. given inner product

Let $s$ be a nonnegative half-integer and $\mathscr P_s$ be the space of complex polynomials $p(z)$ of degree at most $2s$ in the formal variable $z \in \Bbb C$, equipped with the sesquilinear product ...
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What does it mean for a Hermitian sesquilinear form to be 'anisotropic'?

I've seen some references to a 'Hermitian and anisotropic sesquilinear form' with no definition of anisotropy given. According to this Wikipedia page, a quadratic form is anisotropic if $q(v)=0\...
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Cauchy-Schwarz inequality for hermitian quadratic forms

Let $V$ be a $\mathbb C-$vector space and let $\Phi$ be a hermitian quadratic form on $V$. Assume that $\Phi$ is positive definite, i. e., $\Phi(x)>0$ for all nonzero vectors $x\in V$. Let $\varphi$...
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Find a linear operator $A$ associated with the form $\mathfrak{t}$ where $\mathfrak{t}[f,g]=f(c)\overline{g(c)}$

My question is on the title , Let $\mathfrak{t}$ be densely defined form on $\mathcal{H}$. The operator $A$ associated with the form $\mathfrak{t}$ is defined by $Ax=u_x$ for $x\in D(A)$, where $$ D(A)...
Oğuzhan Kılıç's user avatar
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$B(H)$-valued measure: show that a sesquilinear form is bounded

Consider the following fragment from Paulsen's book "Completely bounded maps and operator algebras": To obtain the bounded operator $\phi_E(f)\in B(H)$, we must show that the sesquilinear ...
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Sesquilinear form

Let $V$ be a vector space of finite dimension on $\mathbb C$, $B$ a basis of $V$ , and $f, g$ two sesquilinear forms. If $B$ is an orthonormal basis for $f$ and for $g$ then $f=g$. I want to know if ...
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Regarding Lax-Milgram theorem (Operator theory)

Let $H$ be a Hilbert space, $B\colon H\times H\to C$ a sesquilinear form. Assume that there's $0 \leq M <\infty$ such that for all $x,y\in H$: $|B(x,y)| \leq M\|x\| \|y\|$. A. Show that there is a ...
Mat999's user avatar
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Congruently and Hermitely diagonalisable matrices

It is known that a a matrix is diagonalisable (by the similarity equivalence relation) if and only if there exists a basis of eigenvectors. A typical course in linear algebra then gives two additional ...
AlexInorbit's user avatar
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To prove a sufficient and necessary condition of a quadratic form to equal to zero

The question is the following: Let X a vector space and $B: V \times V \rightarrow K$ a positive semi-definite form. Show that $q_B(y)=0 \Longleftrightarrow B(x,y)=0$, $\forall$ $x$ $\in$ $V$, where $\...
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Bilinear forms and polarization identity

Let $H$ be a Hilbert space and $\sigma: H \times H \to \mathbb{C}$ a sesquilinear form (linear in the first variable, anti-linear in the second variable). Then it is well-known that the following ...
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Symmetric Bilinear Form/ Hermitian Form unique Matrix representation

Given $\mathbb{K} = \mathbb{R} $ or $ \mathbb{C}$ I have $\varphi:\mathbb{K^n}\times\mathbb{K^n}\rightarrow \mathbb{K}$ a symmetric Bilinear Form or complex Hermitian Form/Symmetric sesquilinear form ...
Wolf's user avatar
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What type of matrix guarantees positive-definiteness in complex inner product?

It can be shown that any (complex) inner product on $V=\mathbb{C}^n$ can be written as $\langle v,w\rangle=\bar v^{T}Aw$ for some $n\times n$ matrix $A$. Conversely suppose we consider $F:V\times V\to\...
UnsinkableSam's user avatar
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$H_1$ and $H_2$ are pre-Hilbert Hausdorff space if and only if $H_1 \times H_2$ is pre-Hilbert Hausdorff space

Let $H_1:=(H_1,B_1)$ and $H_2:=(H_2,B_2)$ are two pre-Hilbert spaces. Remembering that 'pre-Hilbert' means that $B_1$ and $B_2$ are non-negative sesquilinear forms in $H_1$ and $H_2$, respectively. ...
Guilherme's user avatar
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3 votes
1 answer
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Show that the sesquilinear form is bounded

Consider the sesquilinear form $$B(f,g)=\int_0^1\bigg(\int_0^xf(t)^*dt\bigg)\bigg(\int_0^xg(t)dt\bigg)dx$$ in $L^2(0,1)$. Show that it is bounded. Let $q(f)=B(f,f)$ be the associated quadratic form. ...
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Sesqulinear forms and weak convergence

Let $a[x,y]$ be a densely defined, symmetric sequilinear form with domain $D(a)$ on a complex Hilbert space. Suppose that $x, y \in D(a)$ and there is a sequence $x_n \in D(a)$ such that $x_n$ weakly ...
Brian Lins's user avatar
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Norm of sesquilinear form bounded by norm of associated quadratic form

I have the following question from Teschl's "Mathematical Methods in Quantum Mechanics": A sesquilinear form is called bounded if $$\|s\|=\sup_{\|f\|=\|g\|=1}|s(f,g)|$$ is finite. Similarly,...
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Hoffman Kunze 9.2 Exercise 6

Problem: Call the form f (left) non-degenerate if $f(\alpha,\beta)=0 \; \forall \beta \implies \alpha =0$. Prove that f is non-degenerate if and only if its associated linear operator $T$ is non-...
Divide1918's user avatar
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Optimize sum of the square of hermitian forms

For M different hermitian forms $h_i(\vec{c}_0,\vec{c}_1)$, where $\vec{c}_j$ are complex vectors in $\mathbb{C}^N$, I want to calculate $$\min_{\vec{c} \in \mathbb{C}^N\backslash\{\vec{0}\}} \frac{\...
Nonabelian's user avatar
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An example of a not positive definite sesquilinear form

Let $R\left(\mathbb{R}/\mathbb{Z}\right)$ denote the space of Riemann integrable periodic functions on $\left[0,1\right]$. Let $<\cdot,\cdot>: R\left(\mathbb{R}/\mathbb{Z}\right)\rightarrow\...
mathgirl796's user avatar
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Generalization of sesquilinear form? [closed]

If a sesquilinear form is a generalization of a bilinear form, what is a generalization of a sesquilinear form ? I don't like this reasoning There’s no need for a generalization of a sesquilinear ...
Jack's user avatar
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Confusion with inner product linearity in complex

For a sesquilinear form, I have encountered two sources that say completely opposite things. During my lecture, my teacher said $\langle \alpha v|w\rangle=\alpha^*\langle v|w\rangle$ but in Wikipedia ...
UnsinkableSam's user avatar
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3 answers
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Sesquilinear form with an operator

Suppose I have an operator $T$ such that $$\forall v\in V,\langle v|Tv\rangle=\langle Tv|v\rangle$$ (where the inner product is defined on complex rather than real) Will it necessarily imply that $$\...
UnsinkableSam's user avatar
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1 answer
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How do I know if a matrix is diagonalizable by congruence?

Let V be a unitary space over $\mathbb{C}^n$ with an orthonormal basis $K$. Let $A: V \times V \rightarrow \mathbb{C}$ be a sesquilinear form. If there exists a basis $B$ in $V$ such that $[A]_B$ is ...
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Showing $x \mapsto B(x,y), y\mapsto B(x,y)$ continuous $\implies B(x,y)\le c\|x\|\|y\|$ for all $x,y\in H$, where $B:H\to H$ sesquilinear, $H$ Hilbert

Let $\mathscr{H}$ be a Hilbert space and $B: \mathscr{H} \times \mathscr{H} \to \mathbb{K}$ be a sesquilinear form, and $\mathbb{K}$ denote $\mathbb{R}$ or $\mathbb{C}$. Show that 1. implies 2., ...
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Proving existence of basis such that the matrix of $f$ is given by $\begin{pmatrix} A&0\\0&0 \end{pmatrix}$

Consider a sesquilinear form $f$ over a finite dimensional vectorspace $V$. Show that the is a basis of $V$ such that the matrix of $f$ is given by $$A_{f} = \begin{pmatrix} A_g &0\\0&0 \...
MyWorld's user avatar
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How does the hermitian sesquilinear form look like?

How does the hermitian sesquilinear form look like? I mean some concrete example. I only found the abstract definitions but I would like to try diagonalize such forms or look at its signature, see ...
Leif's user avatar
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2 answers
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What is a sesquilinear form?

Can anyone please make me understand in easy language what is a sesquilinear form? How could it be used on vector spaces over the field of complex numbers? My Knowledge : Basic knowledge of vector ...
cmi's user avatar
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How to get an $L^1$ from a sesquilinear form?

Let $V$ be a complex vector space. By an $\infty$-sesquilinear form, I mean a sesquilinear form that may take valued in $\mathbb{C} \cup \infty$. If $X$ is a space then, in some sense, $\infty$-...
Kyle Austin's user avatar
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1 answer
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Sesquilinear forms - How does positiveness imply hermitianity?

In my mathematical methods for physics course notes I find this: A positive sesquilinear form is nondegenerate and Hermitian The first statement is trivial: a ...
Jeffrey Lebowski's user avatar
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Limit of sesquilinear forms is a sesquilinear form

Suppose $P_n$ is a monotone sequences of orthogonal projections in a complex Hilbert space $\mathcal{H}$. I want to show that the limit of the sesquilinear forms defined by: $\Gamma_n(x,y)=(x,P_n y)$ ...
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On sesquilinear forms

I was trying to solve some exercises related to sesquilinear forms: Let $V$ be a $\mathbb C$-vector space. Prove that the set $\mathcal{S}(V)$ of sesquilinear forms on $V$ is a vector subspace of ...
Felipe's user avatar
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Polarization of quadratic form yields sesquilinear form

How does polarisation of any quadratic form $Q: V \to \mathbb{C}$ on a complex vector space $V$ yields a sesquilinear form?
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Sesquilinear Forms: Cauchy-Schwarz

This thread is related: Parallelogram Given a Hilbert space $\mathcal{H}$. Consider a quadratic form: $$q:\mathcal{H}\to\mathbb{C}:\quad q[\lambda\varphi]=|\lambda|^2q[\varphi]$$ Suppose it ...
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Sesquilinear Forms: Hamiltonian (II)

Given a Hilbert space $\mathcal{H}$. Consider a positive form: $$s:\mathcal{D}\to\mathcal{H}:\quad s(\varphi,\varphi)\geq0$$ Introduce its form space: $$\mathcal{H}_s:=\mathcal{D}:\quad\langle\...
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Sesquilinear Forms: Hamiltonian (I)

Given a Hilbert space $\mathcal{H}$. Consider a dense positive form: $$s:\mathcal{D}\times\mathcal{D}\to\mathbb{C}:\quad s(\varphi,\varphi)\geq0\quad(\overline{\mathcal{D}}=\mathcal{H})$$ Construct ...
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