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 do different inner products give different angles?

I know that for each inner product $\langle , \rangle_{A}$ on $\mathbb{R}^n$, there is an associated positive definite symmetric matrix $A$ so that $\langle x,y \rangle = x^{T}Ay$. I was wondering if ...
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Find angle between 2 vectors (inner products)

For complex vector spaces, i.e. vector spaces with scalars from the field $C $of complex numbers, inner products must have slightly different properties. To see why, consider the following vectors ...
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Prove $(\sum\limits_k a_k b_k)^2 \leq \sum\limits_k b_k a_k^2 \sum\limits_k b_k$ using Cauchy Schwarz

Let $a,b$ be two $n$ dimensional vectors, we want to show that $(\sum\limits_k a_k b_k)^2 \leq \sum\limits_k b_k a_k^2 \sum\limits_k b_k$ Recall the Cauchy Schwarz inequality is given as $|\langle ...
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How would one prove that the map $\langle \cdot, \cdot \rangle: (f,g) \mapsto \int_{a}^{b}f(t)g(t)dt$ is an inner product?

How would one prove that the map $\langle \cdot, \cdot \rangle: (f,g) \mapsto \int_{a}^{b}f(t)g(t)dt$ is an inner product? We were taught to use this function for inner product related questions, but ...
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Forming the orthogonal space is just a special case of forming a polar set?

Let $M\subset X$ be a subspace and define the polar set $M^{\circ}:=\{x^{\ast}\in X^{\ast}:|\langle u,x^{\ast}\rangle|\le 1\forall u\in M\}$ and $M^{\perp}:=\{x^{\ast}\in X^{\ast}:\langle ...
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Orthogonality in Hilbert Spaces

For the sake of concreteness, let's say that our Hilbert space is the beloved $\mathscr L^2(\Bbb R)$. Suppose that we have $\psi,\phi\in\mathscr L^2(\Bbb R)$, what's the intuitive meaning to a ...
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Counterexample of triangle inequality property for a Real Euclidean Space

According to Apostol's Calculus Vol. II, Equality holds for the triangle inequality in the following scenarios: Let $\mathbb{x}$ and $\mathbb{y}$ be two vectors. Then equality holds if $\mathbb{x} ...
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Why are inner product spaces only defined on $\Bbb R$ or $\Bbb C$?

A vector space $V$ makes sense over any field $F$, or even a division ring. So why does adding an inner product suddenly not make sense without taking the $F=\Bbb R$ or $\Bbb C$? What are the primary ...
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Do Algebraic Extensions have Inner Products?

Given that Algebraic Extensions are vector spaces over fields is it possible to define an inner product?
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if $\langle v , w \rangle = \sqrt {\langle v,v \rangle \langle w, w \rangle}$ then $v=kw$, $k \in \mathbb R$.

Inspired by an earlier question. It is clear to see that if $v=kw$ then $\langle v , w \rangle = \sqrt {\langle v,v \rangle \langle w, w \rangle}$, but why is the other direction also true? This ...
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$|u+w| = |u| + |w|$ iff $\langle u,w \rangle =0$.

I was asked what needs to hold such that $|u+w|=|u|+|w|$. Where $u,w \in \mathbb R^n$. Well, first notice that if $|u+w|=|u|+|w|$ then $|u+w|^2 = |u|^2 + 2|u||w|+|w|^2$ if we go by the definition ...
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Inner product on quantized enveloping algebra

I have a question about a procedure, described in section $2.1.5$ of "Quantum bounded symmetric domains". Here the author describes how to introduce an inner product on $U_q(\mathfrak{g})$. Therefore ...
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Showing two norms on $\mathbb{R}^n$ are dual

I am having trouble showing the following result. If $A$ is a positive definite matrix, then the norms (on $\mathbb{R}^n$) $\|x\|_A:= \sqrt{x^\top A x}$ and $\|y\|_{A^{-1}}:= \sqrt{y^\top A^{-1} ...
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Is it possible to compute the volume of a cone on a inner product space?

This is a matter of curiosity for me. Volumes are often compute using triple integration. But is it possible to compute volumes on a vector space with an inner product defined on that vector space?
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The probability of measuring the control qubit in zero in a quantum circuit

I’m working on an assignment where I have to solve some questions about a quantum circuit. In particular, I have a quantum circuit with three qubits: $|0\rangle$(referenced to as the control qubit), ...
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What does “canonical” mean in vector space?

I was watching this video: https://www.youtube.com/watch?v=RDkwklFGMfo And the professor is talking about the inner product... then he brings up the "canonical" representation of the inner product in ...
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Orthogonal projection by standard norm of and a non-standard norm in $\mathbb{R}^2$

Suppose a closed convex set $S\subset \mathbb{R}^2$ is given by by the convex hull of $(0,1) (-1,1), (-1,0), (1,0)$ and a continuous, convex and decreasing curve $F$ linking $(1,0)$ and $(0,1)$, ...
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Is there only one way to define a norm from an inner product?

Given an inner product $\langle,\rangle$, we can define a norm by $||x|| = \langle x,x \rangle^{\frac{1}{2}}$. My question is, are there other ways to derive a norm from an inner product space and if ...
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P is an orthogonal projector if and only if $P^2 = P$ and $AP$ is symmetric

I'm given that the inner product in a linear subspace $V \subset R^n$ is defined as $<X,Y>_V = X'AY$ where matrix $A$ is positive definite. I need to show that $P$ is an orthogonal projector if ...
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For a regular parametrised plane curve $\alpha$, show that $\langle \alpha''(t),n(t)\rangle =- \langle \alpha'(t),n'(t)\rangle$

When I was proving some properties of regular parametrised plane curve $\alpha:I\to R^2$ which has a normal vector $n(t)$, I encountered the need to prove the following: $$\langle ...
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all possible inner products in $\mathbb R^2$

Suppose $\langle., .\rangle: \mathbb R^2\times \mathbb R^2\to \mathbb R$ is an inner product. What would be all possible function forms of the inner products, i.e. would all of them have the forms ...
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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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Proving that $\langle e_i, e_j \rangle = \delta_{ij}$

Let $(V, \langle \cdot,\cdot \rangle)$ be a real or complex inner product space. Let $\{x_1, ..., x_n\}$ be a linearly independent set. Define: $$e_1 = \frac{x_1}{||x_1||}, z_k = x_k - ...
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maximum vector in $\Bbb R^n$ which any two of them has dot product less than 0

I'm thinking of in $\Bbb R^n$, at most how many vectors can we have any two of them's dot product are less than 0 In $\Bbb R^3$, I guess that there are at most $4$ vector but I can't prove it.(at ...
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Why is the inner product of a quadratic form a quadratic form?

I was going through a derivation of the second derivative of the $\log \det X$ where $X$ is symmetric positive definite, I noticed that despite the second order approximation of log det is written as: ...
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Intuition for orthogonality in infinite dimensions

I'm trying to explain orthogonality in inner product function spaces (e.g. Hilbert spaces) intuitively. As main expample, take the $L^2$ inner product given by $$<f,g>_{L^2(I)}:=\int_I ...
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Does the inner product $\langle \cdot, \cdot \rangle$ induce any other norms other than the 2 norm?

In the lecture my professor wrote that the standard inner product on $R^n$ is given by $\langle x, y \rangle = x^Ty = \sum\limits_{i=1}^n x_i y_i$ which induces a norm $\sqrt{\langle x,x \rangle} ...
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The inner product spaces and linearity in probability

Consider a class of inner product spaces $$\langle \cdot,\cdot\rangle_{{\lambda}\in \Lambda}: R^n\times R^n\to R$$ parameterized by $\lambda \in \Lambda=\Delta(\{w_1,....,w_n\})$, the set of all ...
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Solving Inner Product Equations

I'm trying to solve an exercise from Cheney's Analysis for Applied Mathematics. Let $X$ be a normed linear space with $a,b,c\in X$ taken as fixed vectors, and consider the equation $x+\langle x, ...
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Explain why, if $x \in F$, then $x \in {F}^{\perp\perp}$

As far as the definition of orthogonality ${F}^{\perp} \!$ Is ${y \in E, \forall f \in F, (f|y)=0}.$ Let be $x \in F$, We have by definition of ${F}^{\perp} \!: \forall y \in {F}^{\perp} \!, ...
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Is it always possible to find a vector perpendicular to two given vectors in a general inner product space?

In short, given an inner product space $X$ , for any $x,y \in X$ does there always exist a nontrivial $z \in X$ so that $<x,z> = 0$ and $<y,z> = 0$ ?
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Proving $\langle HX, Y \rangle = \langle X, HY \rangle$ for Hermitian matrices

Let $X, Y \in \mathbb{C}^n$. Let $\langle \cdot, \cdot \rangle$ be the standard Hermitian innerproduct, defined as $\langle X, Y \rangle = \overline{X}^T Y$. I want to prove the following assertion: ...
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Proof that a $L^2$ function space defined on the reals is closed under addition

Here's my attempt: $$L^2$$ is closed under addition. I need to show that $$ \left(\int_a^b |f(t)|^2 \, dt\right)^{\frac{1}{2}}<\infty$$ and $$\left(\int_a^b |g(t)|^2 \, dt ...
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Why are inner products defined to be linear in the first argument only?

It seems to me that if the base field is the real numbers, then we have linearity in both arguments i.e. $\langle u + v, w + z\rangle = \langle u,w\rangle + \langle u,z\rangle + \langle v,w\rangle + ...
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what is the difference between a Hermitian inner product and an inner product?

Are there any difference? Or is the Hermitian inner product just a special case of an inner product on a complex space?
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Showing equivalence of inner products

Show that $\mathcal{A}$ defined by (3) is symmetric, (self-adjoint), on $C_0^2([0,L])$ i.e. show $\langle\mathcal{A}f,g\rangle=\langle f,\mathcal{A}g\rangle,$ for all $f,g \in C_0^2([0,L])$ Hint: ...
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Intuitive meaning of orthogonality of linear transformations w.r.t Frobenius (H-S) inner product

Assume $\langle A,B \rangle = 0$, where $A,B$ are $n \times n$ matrices and $\langle , \rangle$ is the Frobenius inner product. (Also known as the Hilbert–Schmidt inner product). What does it mean ...
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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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If $d(x,z)=d(x,y)+d(y,z)$, then $z-x=t(y-x)$.

Let $x,y,z$ three differents points in a vector space $E$, endowed with inner product. If $d(x,z)=d(x,y)+d(y,z)$, then $z-x=t(y-x)^{(*)}$, with $t\geq 1$. My approach: Let $E$ a vector space with ...
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If $\mathrm{min}_{u\in U}||v-u||=||v-u_0||$, then $v-u_0\in U^{\perp}$

Problem Let $U$ be a subspace of an inner product space V (here V may not be finite dimensional). Fix $v\in V$. If there exists $u_0 \in U$ such that $\mathrm{min}_{u\in U}||v-u||=||v-u_0||$, then ...
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Do any books or articles develop basic Euclidean geometry from the perspective of “inner product affine spaces”?

Definitions. By a vector space, I simply mean an $\mathbb{R}$-module. By an affine space, I mean a vector space $X$ (the "translation space") together with a set $P$ (of "points"), together with an ...
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$\|x\|=\|y\|$ implies that $\langle x+y,x-y\rangle=0$.

Kreyszig - Introduction to Functional Analysis - Page 135 Exercise 4. If an inner product space $X$ is real, show that the condition $\|x\|=\|y\|$ implies that $\langle x+y,x-y\rangle=0$. Okay ...
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Prove that $\sum_{i,j} \langle v_i, v_j \rangle \langle w_i, w_j \rangle \geq 0$

Let $v_1 \dots v_n, w_1 \dots w_n \in H$ an inner product space. I am trying (unsuccesfully) to show that $$ \sum_{i,j=1}^n \langle v_i, v_j \rangle \langle w_i, w_j \rangle \geq 0 .$$ Any hints?
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A vector in a function space

Suppose we let $$L^{p=2}(D,\mathbb{R})$$ denote a set of real functions on a domain D such that if $$\mathbf{a} \in L^{p=2}(D,\mathbb{R})$$ then we have $$\int_{D} \left | a(t) \right ...
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triangular inequality

If we write $||a+b||\leq||a||+||b||$ explicitly in $\mathbb{R}^n$ it is $\sqrt{\sum^n_1(a_i+b_i)^2}\leq \sqrt{\sum^n_1(a_i)^2}+\sqrt{\sum^n_1(b_i)^2}$ how can it be if ...
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What exactly is an integral kernel?

I am not sure if I have seen integral transforms in the right way, but given a transform like Fourier transform - it's actually a basis transformation right ? $$ F(y) = \int K(x,y) f(x) \text{d}x $$ ...
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Riesz representative of gradient of $f(u) = u^*u$ in different inner products

This is a seeming "paradox" that has been bothering me for some time, as it (or other situations like it) show up often when computing gradients for numerical optimization on complex vector spaces. ...
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Dot Product Derviation

The dot product or inner product in Euclidean Space $A\cdot B$ has two definitions: Algebraically defined as: $$A \cdot B = \sum_{i=1}^{n}A_i \cdot B_i=A_1B_1 + A_2B_2 ... A_nB_n$$ Geometrically ...
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Completeness of metric, normed and inner product spaces

For a metric space to be complete, it needs to have all cauchy sequences converge in the metric. 1) For a normed space to be complete, does it need Cauchy sequences to converge in the norm or in the ...