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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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Find the adjoint under the inner product $\langle f, g \rangle = \int_0^1 f(t)g(t)t \ dt$ of $\mathcal{L}(f)(t) = \frac{d^2 f}{dt^2} + f$.

Find the adjoint under the inner product $\langle f, g \rangle = \int_0^1 f(t)g(t)t \ dt$ of $\mathcal{L}(f)(t) = \dfrac{d^2 f}{dt^2} + f$ with $f(0) = 0$ and $f'(1) = 0$. Note: The weight ...
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Defining an inner product by means of a multilinear functional

Let $V$ be a complex vector space and an anti-linear involution $J:V \rightarrow V$ (this means that $J^2 = I$ and if $\lambda \in \mathbb{C}$ and $x, y \in V$ we have $J(\lambda x + y) = \overline{\...
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Prob. 10, Sec. 3.2 in Erwin Kreyszig's “Introductory functional analysis with applications”

Here is Prob. 10 in the Problems after Sec. 3.2 in Introductory Functional Analysis With Applications by Erwin Kreyszig: ... Let $T \colon X \to X$ be a bounded linear operator on a complex inner ...
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Equivalence of hermitian forms under subgroups of $\textrm{GL}_n(\mathbb{C})$

Let $X \in \textrm{GL}_n(\mathbb{C})$ be a hermitian matrix ($\space ^t \overline{X} = X$). For another hermitian matrix $Y$, let's say that $X \sim Y$ if there exists a $g \in \textrm{GL}_n(\mathbb{...
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Is it possible to define an inner product to an arbitrary field?

I've been trying to find the most general definition of an inner product space. Every definition I've found is either to $\mathbb{R}$ or to $\mathbb{C}$. Is it possible to define an inner product to ...
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Inner product space of measures

Let $(X,\Sigma)$ be measurable space and $\mu_1,\mu_2,\dots$ set of finite measures on $X$ such that $\mu_i \perp \mu_j$ for $i\neq j$. Now we can consider space of measures: $$ \mathcal{M} = \left\{ ...
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An inner product on the dual space of a non-complete inner product space?

As is well known, for any Hilbert space $V$, there is a natural inner product on the continuous dual. (the space of all continuous linear functionals). Is there a way to endow an inner product on ...
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Geometric interpretation of $x_1^2y_1^2+x_2^2y_2^2+x_3^2y_3^2+\dots$

Say $x$ and $y$ are two $L_2$ unit vectors of size $n$. In that case the inner product: $$x_1y_1+x_2y_2+x_3y_3+\dots+x_ny_n$$ Is the cosine of the angle between them. For an application I was ...
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Duality pairing and difference with inner product in Hilbert spaces

My question is an extension to the post How is the acting of $H^{-1}$ on $H^1_0$ defined?. Here duality pairings were discussed and even given explicit examples. Let $U$ and $V$ be Hilbert spaces ...
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Divergence of Petersson inner product

Consider a complex number $z = x + i y$ and functions $f,g : \mathbb{H} \to \mathbb{C}$ (where $\mathbb{H}$ is the upper half-plane i.e. complex numbers whose imaginary part is greater equal to zero). ...
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Is the natural norm on the exterior algebra submultiplicative?

For an inner product space $(V,\langle \cdot,\cdot\rangle)$ the exterior algebra $\Lambda V$ inherits an inner product, which satisfies $\langle a_1\wedge \dots \wedge a_n,b_1\wedge\dots\wedge b_n\...
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Does projections to orthogonal summands closed $\implies$ subspace closed?

Let $X$ be a inner product space, with orthogonal decomposition $X=V \oplus W$. Give $X$ the topology induced by the norm induced by the inner product. Let $E\subset X$ be a subspace such that the ...
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Are projection and norm enough to define an inner product?

Given an inner product, one can define a projection and a norm. Can we do the opposite? That is, suppose we have: a complex vector space V a norm $|V|^2 : V \rightarrow \mathbb{R}$ such that: is ...
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Progressed : Convergence problem in Hilbert Space and necessity of inner product

******** PROGRESS : so thanks to Ian's great comment I can get by the proof and that completeness is necessary but I need to know does this hold for general Banach spaces that are not Hilbert spaces? ...
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If for every operator represented by $A$ w.r.t to a basis $\mathcal{B}$, the matrix representation of $T^*$ is $A^*$, then $\mathcal{B}$ is orthogonal

Let $V$ be a finite-dimensional inner product space. Assume that for every linear operator $T$, represented by $A$ w.r.t to a basis $\mathcal{B}$, the matrix representation of the adjoint w.r.t to $\...
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Orthonormal basis of polynomials

I am trying to find an orthonormal basis of the vector space $P^{3}(t)$ with an inner product defined by $$\langle f, g \rangle = \int_0^1f(t)g(t)dt$$ by applying the gram schmit alogorotin to ${(...
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Application of Cauchy-Schwarz with Sobolev norms

I'm working through the problems in the initial value formulation chapter in Wald's General Relativity. A short summary of the problem. I have to show that $$\sup_{x\in A}|f(x)|\le C||f||_{A,k}$$ ...
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Orthonormal basis Parsevals identity.

Let $O={u_1,...,u_k}$ be an orthonormal set in $V$. Prove that $O$ is an orthonormal basis if and only if Parseval's identity holds for all $v,w \in V$ i.e if and only if $$\langle v,w\rangle=\sum_{...
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Homework: Second derivative of $\langle Ax, x \rangle$

So let $A \in M_{n}$ and define $f: \mathbb{R}^n \to \mathbb{R}$ by $f(x) = \langle Ax, x \rangle $. Find f' and f''. After some work, I found the first derivative to be $f'(x)(v) = \langle Ax, v \...
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Inner Product as Weighted Average

Let $\zeta=e^{2\pi i/n}$, where $n\geq3$. Let $||\cdot||$ be the norm induced by the complex inner product $\langle\cdot,\cdot\rangle$. Then $$\langle x,y\rangle=\frac{1}{n}\sum_{k=1}^{n}||x+\zeta^...
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Integral of inner product of $\langle x, A(t) \dot{x}\rangle$

Let $A(t)$ be uniformly positive definite for any $t$, and $x(t)$ is a vector, and $x(0)$ is a finite. since $\int_0^t x(\tau)\dot{x}(\tau)\,\mathrm d\tau = \frac{1}{2}[x(t)^2-x(0)^2]$ is finite, I ...
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$\| M - A \|_F^2 \geq \sum_{h = l+1}^k \lambda_h$

Let $A \in M_{n,p}(\mathbb{R})$ such that the rank of $A$ is $k \leq \min(n,p)$. Moreover let $M \in M_{n,p}(\mathbb{R})$ be a matrix of rank $l$ with $l+1 \leq k$. We denote $(\lambda_i)_{i \leq k}$ ...
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Riemaniann metric problem in K&N's book.

I'm reading this book by S. Kobayashi and K. Nomizu, "Foundations of Differential Geometry, Vol.1" and I have a problem in the proof of this lemma at the page 170: And the proof gose like this: My ...
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Prove that the polar cone $K^\circ = \{v\in\mathbb{R}^n:Av\leq 0\}$ where $A$ is the matrix that has $x^i$ as its $i$-th row for ($i=1,\dots\,m$).

Let $x^1,\dots,x^m$ be $m$ vectors in $\mathbb{R}^n$ and let $K = \text{cone}(\{ x^1,\ldots, x^m \})$ be the cone generated by the set $\{ x^1,\ldots, x^m \}$. (Superscripts are indices not powers.) ...
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True/false in linear algebra exam : inner product spaces

I found a true/false exam. Here is the exercise (other question are here: 2) Let $q(x)=x^7+1$. Set $W=\{p(x)\in \mathbb R_n[X]\mid p(0)=0\}$. Let $r(x)\in W$. Suppose $$\int_0^1 (q(x)-f(x)p(x)dx=0,$$ ...
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Inner product in module theory

Consider an $R$-module $M$, for a $R$ a commutative ring with identity. Of course one could define the notion of inner nondegenerate* symmetric product $(-,-)$ in the same fashion of vector spaces. ...
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If an inner product space has an “orthonormal” Hamel basis, then does any subspace of it has an orthonormal Hamel bais?

Here, the concept of "orthonormal Hamel basis" of an innper product space $V$ is a family of orthonormal family $\{e_i\}_{i\in I}$ (may be uncountable) such that any $x\in V$ is a finite combination ...
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“Projecting” one convex set onto another (though actually, it's just a translation).

In linear algebra, we learn that we can project a vector $x \in \mathbb{R}^n$ onto a linear subspace $A \subseteq \mathbb{R}^n$. I have hunch that this can be generalized considerably. In particular, ...
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Strong convexity and the Legendre transform

Suppose that I have a strongly convex function $f(\mathbf{x}): \mathbb{R}^m \rightarrow \mathbb{R}$. Is the Legendre transform of this function also strongly convex? As far as I can tell, strict ...
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Conditional Expectation as Inner Product

Hansen and Richard (1987), Page 592, introduces a conditional counterpart to an inner product and also defines a conditional norm. They define for random variables X and Y and $\sigma$-field G $<X,...
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Construction of a norm in Lp such that it derives from an inner product

In $L^p((X, \mu), \mathbb{R})$, $p \geq 2$, it is possible to define the norm (considering that $\mu(X) = 1$.) $$||f||_2 = \left( \int_X f(x)^2d \mu(x) \right)^{1/2}$$ which comes from the ...
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Prove that $X$ is not Hilbert

Let $X$ be vector space of real-valued function with continuous derivative on $[0,1]$. Define an inner product on $X$ by $$\langle x, y \rangle = x(0)y(0) + \int_0^1 x^\prime (t) y^\prime (t) dt.$$ ...
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Inner-product on skew-hermitian matrices

Let $$\mathfrak{u}(n)=\{X\in M(n,\Bbb C):X+X^*=0\}$$ where $X^*$ is the conjugate transpose. Then, $\mathfrak{u}(n)$ is a real vector space. Problem. Show that $\langle X,Y\rangle=\...
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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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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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Inner product for dual space

Suppose we have a Hilbert space $H$. Is there any explicit expression for the inner product on $H^*$ without resorting to Riesz representation theorem? I am NOT looking for one that uses the ...
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A simple way to solve that inner product and functional question

Let $V$ be the polynomials space over $R$ of degree less than 3 with inner product $$\langle f,g\rangle= \int^{1}_{0} f(x)g(x) dx $$ if $x \in R$, compute $g_{x}$ such that $\langle f,g_{x}\rangle =...
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Do I have the correct mental map for adjoint operators for inner product spaces?

Let $X$, $Y$ be finite dimensional inner product spaces, let $A: X \to Y$ be a linear operator, let $A^*: Y \to X$ be the adjoint operator to the linear operator, defined using $<y, Ax>_Y = <...
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Is the inverse of a bijective connectedness preserving map , on a complete real inner product space , also connectedness preserving?

Let $X$ be a complete real inner-product space and $f:X \to X$ be a bijection which maps connected sets to connected sets ; then is it necessarily true that $f^{-1}$ also maps connected sets to ...
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Pureness of Vector States

How does one show that irreducibility is equivalent to a vector state being pure? In what follows I will fill in the details of the question: Let $\mathcal{H}$ denote a Hilbert space and let $\...
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Find all $ p \ge 1 $ for which the Hölder norm $\|\cdot\|_p $ is generated by a scalar product.

Find all $ p \ge 1 $for which the Hölder norm $$ \|x\|_p := \left(\sum^{n}_{i=1} |x_i|^p\right)^{\frac{1}{p}} $$ is generated by a scalar product. We know that norm is generated by a scalar product ...
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Inner product exterior algebra

I have to prove that if $V$ is a real vector space ($\dim V=n$) with inner product $(.,.)$ then if we define $$ (v_{1}\wedge v_{2}\wedge\cdots\wedge v_{k},w_{1}\wedge w_{2}\wedge\cdots\wedge w_{k}) =...
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Hermitian positive semi-definite matrix is a Gram matrix

I showed that every Gram matrix, i.e. a $n \times n$ matrix $A$ with $A_{ij} = <x_i,x_j>$ where $x_1,...,x_n$ are vectors in an inner product vector space $V$, is Hermitian and positive semi-...
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Shouldn't 'without loss of generality' be mentioned here?

Regarding the blocked portion my question is what if $x\ne z=y?$ Shouldn't without loss of generality be mentioned in the text?
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Inner spaces - A question about adjoints.

Question: Let $V$ be a complex vector space over $\mathbb{C}$ with inner product $\langle ,\rangle$. Let $E$ be an linear operator on $V$ such that $E^2=E$ with adjoint $E^*$. Show that $E$ is ...
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Sum of inner products in a hilbert space

This question asks to show that the following inner product defines a Hilbert space/is complete: $$\langle f,g\rangle=\sum_{k=0}^{n}\int_0^1 f^{(k)}(t) \overline{ g^{(k)}}(t)dt$$ where $f^{(k)}(t)$ ...
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Existence of n distinct (real) roots of an orthogonal polynomial

I'm trying to get my head around the proof that an orthogonal polynomial ($P_n$ say) has at least n distinct roots. My understanding of the proof http://en.wikipedia.org/wiki/Orthogonal_polynomials#...
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A construction of the Hodge Dual operator

This question about showing that an alternative construction of the Hodge dual operator satisfies to the universal property through which the Hodge dual is usually defined. Let me give the ...
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How to prove that $(C( F), \langle.,.\rangle_2)$ is a pre-Hilbert space?

Let $F$ be a surface. For all continuous functions $f,g \in C(F) $ define $$ \langle f,g\rangle_2 := \int_F f(x)g(x)\, dx $$ I'm struggling to show, that $ (C(F),\langle.,.\rangle_2) $ is a pre-...
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Find $T^*(x,y)$ with the given inner product

Let $T:\mathbb{C}^2 \to \mathbb{C}^2$ be a inner product space, $T(x,y)=(2ix+y, -x-iy)$ and the inner product $<(x_1, x_2), (y_1, y_2)>=4x_1\overline{y_1}+9x_2\overline{y_2}$. Find $T^*(x,y)$ ...