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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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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53 views

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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28 views

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 ...
4
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56 views

Integral of a function with an exponentiated inner product

Let $f:\Bbb R^n\to \Bbb R^n$ be a continuous function such that $\int_{\Bbb R^n}|f(x)|dx\lt\infty$. Let $A$ be a real $n\times n$ invertible matrix and for $x,y\in\Bbb R^n$, let $\langle x,y\rangle$ ...
4
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148 views

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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52 views

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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273 views

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 ...
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32 views

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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+50

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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57 views

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 ...
3
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76 views

Prob. 10, Sec. 3.2 in Erwine 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 Erwine Kreyszig: ... Let $T \colon X \to X$ be a bounded linear operator on a complex ...
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38 views

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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49 views

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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118 views

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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781 views

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 ...
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37 views

Gradient of inner product in Hilbert space

Let $\mathcal{H}$ be a Hilbert space and \begin{align} f&\colon \mathcal{H} \to \mathbb{R}\\ f(x) &= ||x-c||_\mathcal{H} ^2 \end{align} from some constant $c \in \mathbb{H}$ Is the derivative ...
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29 views

Distribution of $\langle A,x\rangle\langle A,y\rangle + \langle B,x\rangle\langle B,y\rangle$ given $\langle x, y\rangle$

Let $A$ and $B$ be independent, normal distributed $N(0,1)$ normalized unit vectors, and let $x$ and $y$ be unit vectors with given inner product $\langle x, y\rangle=u$. Can we write the ...
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39 views

What's the advantage of using bra-ket notation over inner-product notation?

Both notations look pretty similar, and appear similar when undergo algebraic operations. Apart from personal taste (aesthetics) concerning commas ($\langle \phantom{\cdot},\phantom{\cdot} \rangle$) ...
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28 views

Extend $(\frac{1}{2}, \frac{i}{2} ,\frac{-1}{2},\frac{-i}{2} )$ to an orthonormal basis for $\mathbb{C}^4$.

Consider $\mathbb{C}^4$ with the standard inner-product$ < , >$. Extend $(\frac{1}{2}, \frac{i}{2} ,\frac{-1}{2},\frac{-i}{2} )$ to an orthonormal basis for $\mathbb{C}^4$. How is this possible ...
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64 views

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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80 views

Inner product exterior algebra

I have to prove that if V is a real vectorial space (dimV=n) with inner product (.,.) then if we define $$ (v_{1}\wedge v_{2}\wedge...\wedge v_{k},w_{1}\wedge w_{2}\wedge...\wedge w_{k}) ...
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42 views

Intuition behind complex inner product

Let $f_n : \mathbb C^n \to \mathbb R^{2n}$ be defined by $$(x_1 + iy_1, x_2 + iy_2, \dots) \mapsto (x_1, y_1, x_2, y_2, \dots )$$ I am having trouble believing that $$ \langle X, Y\rangle_{\mathbb ...
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18 views

The problem of support vector machine - How to minimize $||w||^{2}$ subject to constraints of the form $\alpha w_{1}+\beta w_{2}+b\geq\pm1$

I am studying the subject of support vector machines from an online course. I am given four points and their classification $$ x_{1}=((5,4),+),\, x_{2}=((8,3),+) $$ $$ x_{3}=((3,3),-),\, ...
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48 views

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, ...
2
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62 views

Showing an inner product space is complete

I'm working through Ward Cheneys Analysis for Applications and I'm a bit stuck on this exercise from Section 2.2: Prove that if $M=M^{\perp\perp}$ for every closed linear subspace $M$ in an inner ...
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30 views

Using derivatives at 0 to define an inner product

Can the following define an inner product on a subspace of the set of functions that are infinitely differentiable on $[-R,R]$. If so, do we get a Hilbert space? $$<f, g> = \sum_{n=0}^\infty ...
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41 views

Inner product space or Hilbert space of Quaternionic Functions

In what ways can you define an inner product, $<f,g>$, to create an inner product space or Hilbert space on the set of quaternionic functions $f:\mathbb{H} \rightarrow \mathbb{H}$?
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242 views

Duality pairing and difference with inner product in Hilbert spaces

My question is an extension to the post Acting of a dual pairing in Sobolev Spaces. Here duality pairings were discussed and even given explicit examples. Let $U$ and $V$ be Hilbert spaces such that ...
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30 views

Linear group action over an hermitian space.

Let $\mathrm{h}$ be an non degenerate hermitian form on $\mathbb{C}^2$, and suppose $\mathrm{h}$ is of signature $(1,1)$. Let $A$ and $ B \in \mathrm{SL}(2,\mathbb{C})$ such that $G$ the group ...
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115 views

Weighted $L^p$ spaces and orthogonal families of polynomials

Let $I$ be a closed interval (bounded or not) of $\mathbb{R}$ and let $w\colon I \rightarrow \mathbb{R}$ be a continuous function that is positive inside $I$ and such that for every $n\in ...
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214 views

Is my formula for the change of basis of an inner product correct?

Suppose we have a real or complex finite dimensional vector space $V$ with an inner product $\langle \cdot , \cdot \rangle$ and a basis $B = \{v_1, \dots, v_n\}$. Define the matrix of the inner ...
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458 views

Gram-Schmidt Orthogonalization for subspace of $L^2$

I am a little stuck on the following problem: By using the Gram-Schmidt Orthogonalization, find an orthonormal basis for the subspace of $L^2[0,1]$ spanned by $1,x, x^2, x^3$. OK, so I have defined: ...
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164 views

Different topologies on a normed/inner product space

Given a normed vector space $X$. Let $\mathcal{T}$ represent the topology on $X$ induced by the norm. Define: $A$:={topologies that can make the norm continuous}, $B$:={topologies that can make the ...
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How can we check that for a given norm, we can found an inner product?

Let $$\Bbb C^2=\{w=(z_1,z_2) : z_1,z_2\in\Bbb C\}$$ be the vector space of all ordered pairs of complex numbers. Can we obtain the norm defined on $\Bbb C^2$ by $$||w||=|z_1|+|z_2|$$ from an inner ...
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36 views

Eigenvalue for a conjugate operator.

$\newcommand{\lbrac}[1]{\left( #1 \right)}$ Let $V$ be a complex inner product space, and let $T:V\to V$ be a linear operator over $V$ and $T^*$ its adjoint. Suppose $\lambda$ is an eigenvalue of $T$. ...
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20 views

Show that $\{x\in V| \langle x,e \rangle=0 \forall e\in E\} =\{y\in V ~| ~y\perp w_i, 1\leq i \leq k \}$

Let $E$ be subset of a vector space $V$. Let $B =\{w_1,\dots,w_k\}$ be a basis for $E$. Prove: $E^\perp =\{y\in V | y\perp w_i, 1\leq i \leq k \}$ Is my proof correct? Define two sets: (a) ...
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19 views

Concentration of measure of inner product in Hilbert space?

In the finite dimensional Hilbert space of quantum mechanics (one where all vectors have norm one), is a concentration of measure phenomenon observed with the inner product of any two vectors? That ...
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78 views

Prob. 9, Sec. 3.2 in Erwine Kreyszig's “Introductory Functional Analysis with Applications”

Here is Prob. 9 in the Problems after Sec. 3.2 in Introductory Functional Analysis With Applications by Erwine Kreyszig: Let $V$ be the vector space of all continuous complex-valued functions on ...
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45 views

Characterisation of inner products preserved by an automorphism

Let $V$ be a finite dimensional vector space. Let us call an automorphism $T:V\rightarrow V$ admissible if there exists an inner product $\langle , \rangle$ on $V$ making $T$ an isometry. (You can ...
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Subspace of $L^p(X,\Sigma,\lambda)$

Consider $R$-valued functions in $L^p(X,\Sigma,\lambda)$, where $X=X^1\times X^2$, $\Sigma=\Sigma^1\times \Sigma^2$ and $\lambda=\lambda^1\times \lambda^2$ For given $i$, does the subsapce $M=\{f\in ...
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22 views

Complex euclidean tensor products

Say you have Euclidean vectors $\mathbf{a}=a_i \mathbf{p}_i$ and $\mathbf{b}=b_j \mathbf{q}_j$ in $\mathbb{R}^3$, with bases $\mathbf{p}_i$ and $\mathbf{q}_j$. Then you could use a typical inner ...
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36 views

To show that orthogonal complement of a set A is closed.

To show that orthogonal complement of a set A is closed. My try: I first show that the inner product is a continuous map. Let $X$ be an inner product space. For all $x_1,x_2,y_1,y_2 \in X$, by ...
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66 views

Inner product issues

STATEMENT: Suppose V is a finite dimensional vector space with an inner product $\langle *,*\rangle$. And $\phi :V\rightarrow V^*$ is the isomorphism $\phi(v)=\langle v,*\rangle$ The inner product, ...
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49 views

Maximizing inner product of unit vectors

I am working on a research project and the main algorithm is based on computing the following function: Given a symmetric matrix $A$ and a unit vector $x_0$, compute: $\max\langle Ax,x\rangle : ...
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79 views

Orthogonality in Space of Polynomials of Degree at Most 2

Let $E$ be the space of polynomials of degree at most $2$. On $E$ define $\langle f,g \rangle := f(-1)\overline{g(-1)}+f(0)\overline{g(0)}+f(1)\overline{g(1)}$ for $f,g \in E$. a). Show that this ...
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15 views

name, notation for “block inner product” $X^H Y$

Given a set of $k$ vectors of length $n$, $X = [x_1, \dots, x_k]$ and another set of $l$ vectors of length $n$, $Y = [y_1, \dots, y_l]$, I'd like to to compute the inner product of every combination ...
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21 views

It is always possible to find a norm on an inner product space E which would define the weak convergence in E?

It is always possible to find a norm on an inner product space E which would define the weak convergence in E?
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44 views

Show that $L^2 ([ a, b])$ is the only inner product space among the spaces $L^P([a, b])$.

I know that $L^2 ([ a, b])$ is an inner product space with parallelogram law. But I cannot prove it for $p \neq 2$.
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141 views

Orthonormal set is a Hilbert basis $\iff$ Parseval's identity is true

Let $H$ be a Hilbert space and $\{e_k:k\in \mathbb{Z}\}$ an orthonormal set. Prove that the set is a Hilbert basis if and only if Parseval's identity is true. The direct theorem is almost ...
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99 views

Dimension of image of a skew symmetric map is even

If $A$ is a skew-symmetric linear transformation on a finite-dimensional Euclidean space, then rank $\rho(A)$ of $A$ i.e., the dimension of image of $A$ is even. I am trying for a geometric proof of ...