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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132 views

Generalization of inner product spaces (analogue to uniform spaces/locally convex spaces)

In the following I am going to devise a chart of topological spaces that contains inner product spaces, normed vector spaces, metric spaces and other related spaces. In the end there will be a gap in ...
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127 views

Application of Cauchy-Schwartz 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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255 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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34 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

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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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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113 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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715 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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57 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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51 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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40 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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15 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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47 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, ...
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25 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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38 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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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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113 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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200 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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432 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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157 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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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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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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31 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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58 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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44 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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71 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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13 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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26 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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23 views

Given two dot products with the same vector in a prime finite field of 2 (Galois Field), how can one figure out future dot products?

I've stumbled upon an interesting "rule" derivation for the value of a dot product in $\mathbb{R}^{n}$ like this: Given an arbitrary vector $\vec a \in \mathbb{R}^{n}$ and the values of two dot ...
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87 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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97 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 ...
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34 views

Directional derivative, why is this incorrect?

A function $\vec{F}$ is dependent on the lenght (norm) of $\vec{r}$ ${\vec{F}(\vec{r})=F_{x}(\vec{r})\hat{x}+F_{y}(\vec{r})\hat{y}+F_{z}(\vec{r})\hat{z}} $, in which ...
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28 views

When can we take $A$ somehow out of $\langle x,Ay\rangle$?

Let $A$ be a linear mapping on an inner product space $V$ and $x,y \in V$. What are some cases (as general as possible) can we take $A$ somehow out of $\langle x,Ay\rangle$? (I know when $A = c I$, we ...
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34 views

Is the orthogonal complement of $V=C[0,1]$ is ${0}$ in $L^2[0,1]$

I was trying to think about the orthogonal complement of $C[0,1]$ in L$^2[0,1]$. I thought that it should be $\{0\}$ but I had only little confident with my proof so I'd like to ask you if it's ...
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41 views

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 ...
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63 views

$x \cdot x$ in inner product space is a quadratic form

Given an inner product space with some inner product $\cdot$ , how can I prove that $x \cdot x$ for any vector $x= (x_1,... x_n)$ is a quadratic form in $x_i$? I know how to recover an inner product ...
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96 views

Show linearity of this map

We have the following maps on a complex vector space $V$ $\phi : V \rightarrow \mathbb{C}$ and $g : V^2 \rightarrow \mathbb{C}$ where $\lambda \in \mathbb{C} , x,y,w \in V$. $\phi $ satisfies that ...
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Is it true that $V$ and $V^*$ are naturally isomorphic as finite vector spaces if $V$ is equipped with an inner product?

This is a homework question from my differentiable manifolds class: In general we know that if $\dim V<\infty$ then $V$ and $V^*$ are isomorphic because any two vector spaces with the same ...
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Is it possible to find the norm fuction of a space from an inner product already defined for it?

I'm a noob on the subject of functional analysis. As the title of the question says: Is it possible to find the norm fuction of a space from an inner product already defined for it? e.gr.: Suppose ...
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41 views

Gradient of an Inner Product in a more general Vector Space

I was looking at the following question: Differentiating an Inner Product that was talking about the derivative of an inner product to be: $$ \frac{d}{dt} \langle f, g \rangle = \langle f(t), ...
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21 views

How do you show the connection of reproducing kernels to feature maps?

This question is in the context of Hilbert Reproducing Hilbert Spaces and reproducing Kernels and their relation to feature maps (and machine learning). We have a Hilbert space $\mathcal{F}$ and ...
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104 views

Orthogonal Complement

"Let $\Bbb{V}$ be a vector space with an inner product $<\cdot,\cdot>$, and $S\subset\Bbb{V}$. We define the orthogonal complement of $S$, denoted by $S^{\perp}$, as follows: ...
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17 views

What is the closest self-adjoint (positive) operator to a given operator?

Given an operator $\rho$ on a Hilbert space $H$, is there a notion of nearest self-adjoint (positive) approximation of $\rho$ for a suitable norm? More specifically, does there exist an algebraic ...
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79 views

Is an inner product continuous?

It is an easy question, but i want to make it clear :) Let $(V,\langle -,- \rangle)$ be an inner product space over $\mathbb{K}$. Then, is the inner product $\langle -,- \rangle:V\times V\rightarrow ...
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22 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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57 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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165 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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65 views

Linear Algebra quick Question over inner product space

In an inner product space, not necessarily $\mathbb R^n$, there are vectors $a$ and $b$ such that $||a||\cdot ||b|| < |\langle a,b\rangle| $ Is this never true?