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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Topological space $\nRightarrow$ Metric space $\nRightarrow$ Normed space $\nRightarrow$ Inner product space (Examples)

If I have an inner product space, the hierarchy goes: Inner product space $\Rightarrow$ normed space $\Rightarrow$ metric space $\Rightarrow$ topological space. The reverse, however, is not always ...
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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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Nonexistence of local isometry between equidimensional Riemannian manifolds

Recall that all inner product spaces of the same dimension are isometric. For example, if $(M,\mathrm{g})$ and $(N,\mathrm{h})$ are Riemannian manifolds of the same dimension, then ...
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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 $\langle x,y\rangle$ ...
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orthonormal basis claim (inner product space)

I just need to know if this claim is true or not (at the link). don't need to prove. I know that one direction is true, not sure about the other... Let $T\colon V\to V$ be a linear operator. $B$ ...
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What is the angle between these unit vectors? [closed]

Let $a$ and $b$ be two unit vectors and $p$ is the angle between them. If $a+b$ is also a unit vector, then: $p = \pi/3$ $p = \pi/4$ $p = \pi/2$ $p = 2\pi/3$
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1answer
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Total derivative of inner product.

Let $$F:\Bbb R^n\times\Bbb R^n\to\Bbb R$$ be the function $F(x,y)=\langle Ax,y\rangle$ where $\langle , \rangle$ denotes the standard inner product on $\Bbb R^n$ and $A$ be an $n\times n$ real matrix. ...
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A norm which is symmetric enough is induced by an inner product?

$\newcommand{\<}{\langle} \newcommand{\>}{\rangle} $ It is a fact that for every norm $\| \|$ on a finite dimensional (real) vector space, its isometry group $\text{ISO}(|| \cdot ||)$ is ...
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Inner product space of continuous functions, showing inexistence

Let $E$ be the inner product theorem of the Continuous functions over the interval $[a,b]$, with the inner product $(x,y)=\int_a^b x(t)y(t)dt$. Fix $c\in [a,b]$, and let $f_c\in E^*$ such that ...
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Functionnal analysis: Why $\langle AAx,x\rangle\underset{(*)}{\leq} (\|A\|+m)\langle Ax,x\rangle-\|A\|m ?$

Let $(X,\langle\cdot ,\cdot \rangle)$ an inner vector space and $A\in \mathcal L(X)$ symetric such that $A\geq 0$. I set $m=\inf\{\langle Ax,x\rangle \mid x\in X, \|x\|=1\}$ and thus $A-mI\geq 0$. ...
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Prove that there is an inner product on $\mathbb{R}^2$, given that the associated norm is a p-norm only if p = 2

Prove that there is an inner product on $\mathbb{R}^2$, such that the associated norm is given by: $ \parallel (x,y) \parallel = (|x|^p + |y|^p)^\frac{1}{p}$ where $ p > 0 $ only if $ p = 2 $ So ...
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1answer
37 views

Functional analysis: $\|A\|=\sup\{|\langle Ax,x\rangle|\mid x\in X, \|x\|\leq 1\}$

Let $(X,\langle\cdot ,\cdot\rangle)$ an inner product space and $A\in\mathcal L(X)$. I have to show that $$\|A\|=\sup\{|\langle Ax,x\rangle|\mid x\in X, \|x\|\leq 1\}.$$ The fact that $\|A\|\geq ...
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$\{T(a)| \;T:\mathbb{C}^n\longrightarrow\mathbb{C}^n$ is a $\mathbb{C}$-linear isometry with $\det(T)=1\}=\mathbb{S}^{2n-1}$

Let $a,b\in\mathbb{C}^n$ ($a\neq b$ and $|| a||=||b||$). How can one prove that there is a $\mathbb{C}$-linear isometry $T:\mathbb{C}^n\longrightarrow\mathbb{C}^n$ (that is $T(z)=U\cdot z$ where $U$ ...
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Linear algebra: from the inner product to the dot product

Let $W$ be a $n$-dimensional vector space. What do we mean when we say that a linear transformation $f:W\to \mathbb{R}^n$ carries the inner product on $W$ to the dot product on $\mathbb{R}^n$? Is ...
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How do I show that in a normed space $| (\|x\|-\|y\|) | \leq \|x-y\|$ [duplicate]

How do I show that in a normed space $| (\|x\|-\|y\|) | \leq \|x-y\|$ I need to use normed space axioms but I'm unable to figure it out. I think the scalar axiom and triangle inequality are helpful.
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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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1answer
42 views

inner product space questions

I didn't succeed in solving these two short questions (in the link). In the first one I think I need to work with bases, but I don't know how. In the second question I just don't know how to start. ...
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1answer
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How to generate vectors with given a certain vector and an angle?

Let us start with 3-dimensional case. Let $V_1=(X_1,Y_1,Z_1)$ and an angle $\theta$ be given, how to generate all the other possible vector $V_2=(X_2,Y_2,Z_2)$ such that $<V_1,V_2>=cos\theta$, ...
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Isometry group of a norm is always contained in some Isometry group of an inner product?

$\newcommand{\<}{\langle} \newcommand{\>}{\rangle} $Let $||\cdot||$ be a norm on a finite dimensional real vector space $V$. Does there always exist some inner product $\<,\>$ on $V$ such ...
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1answer
21 views

Inner product axioms

Is there a shortcut to finding out if a particular operation is an inner product? Applying the axioms takes a long time especially when in exams so is there a quick way to find out if the operation is ...
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23 views

Work out the adjoint of $T(x,y) = (y,-x)$

this seems like a simple question but I don't understand it. We define a transformation $T(x,y) = (y,-x)$. We want to work out what the adjoint is. I know the answer: $T^*(x,y) = (-y,x)$ but how? ...
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Confirm my understanding of adjoints

adjoints seem REALLY important and useful so I don't want to move onto the next topic without really understanding them; I have too many a times moved on and been lost because I don't have the ...
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3answers
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Find the adjoint of this non-standard inner product space

I'm really blanking out (a lot of late nights these past 10 weeks). The point of the exercise I'm about to type up is to show that the adjoint structure may possibly change when the inner product ...
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37 views

Projection of $z$ onto $\{x\mid Ax = b\}$

Suppose $A$ is fat(number of columns > number of rows) and full row rank. The projection of $z$ onto $\{x\mid Ax = b\}$ is (affine) $$P(z) = z - A^T(AA^T)^{-1}(Az-b)$$ How to show this? ...
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Is this dual pairing the same as the inner product?

If $(V, \langle \cdot$ , $ \cdot\rangle)$ is an inner product space with dual $V^*$ then there is a natural dual pairing $\langle \cdot$ , $ \cdot \rangle ^*: V^* \times V \rightarrow \mathbb K$ given ...
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56 views

Inner product in Linear algebra

Let $a=\begin{bmatrix} 2 & 3 \\ 1 & -3 \end{bmatrix}$ and $b=\begin{bmatrix} 3 & -1 \\ 0 & 1 \end{bmatrix}$ a) compute $\langle a,b\rangle$ b) Compute $\|a\|$ and $\|b\|$ My ...
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question on projections in linear algebra

Hello all I was given this question for linear algebra class which I am stuck on and would truly appreciate the help: V is a finite-dimensional inner product space with M and N non trivial subspaces. ...
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Let $V$ be an inner product space. Show that if $||x+y||=||x||+||y||$, then $ax=by$ where $a,b$ are non-negative and not both zero.

Let $V$ be an inner product space. Show that if $||x+y||=||x||+||y||$, then $ax=by$ where $a,b$ are non-negative and not both zero. I know that the converse is true. I considered the square of the ...
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If $F$ is an orthonormal basis ,we represent every $x \in X$ as a linear combination consists of finitely many terms ?

If $F$ is an orthonormal basis in an inner product space $X$,can we represent every $x \in X$ as a linear combination consists of finitely many terms ? in Erwin Kreyszig's book says no, but i dont ...
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An equation involving inner products being independent of the inner product space

Let $(X,\langle\cdot\,,\cdot\rangle_X)$ and $(Y,\langle\cdot\,,\cdot\rangle_Y)$ be nonzero inner product spaces over $\mathbb{C}$. I wish to know if the following statement is true. ...
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inner product space problem $(x_n,y_n)\to 0$

If $(y_n)$ is a bounded sequence in an inner product space, and $(x_n)$ is a sequence converging to zero, prove that $(x_n,y_n)\to 0$. Where $(x_n,y_n)$ is the inner product. Since $(y_n)$ is bounded ...
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Finding an inner product given a norm

Let $V=\mathbb{R}^n$. Given a norm $|\cdot |$, find the inner product for which $\langle v,v\rangle=|v|$. So basically I need to find an inner product such that $\forall v\in V. \langle ...
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Proof in inner product [closed]

I am learning for my finale and having issues proving the two statements: $$ \|U\|=∥V∥ \iff\langle U+V,U-V\rangle=0 $$ $$ \|U\|^2=\|V\|^2=\langle U,V\rangle \rightarrow U=V $$ your help will be ...
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Find an inner product that makes a given set of linearly independent vectors orthogonal

I need to find an inner product such that given a set $S$ of linearly independent vectors in a Hilbert space $H$, $S$ will be orthogonal with these product. I thought Gram -Schmidt Process would help ...
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Finding an inner product such that these vectors are orthonormal [closed]

The task goes like this: Find an inner product in $\mathbb{R}^3$ such that the vectors $$a_1=(1,1,1),\qquad a_2=(1,0,1),\qquad a_3=(1,1,0)$$ are orthonormal. How do I find the inner product? ...
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Example 3.10-2 in Kreyszig's functional analysis book: Relation between inner products involving an operator and its matrix

Let $T \colon \mathbb{C}^n \to \mathbb{C}^n$ be a linear operator, which is bounded of course; so the Hilbert adjoint operator $T^* \colon \mathbb{C}^n \to \mathbb{C}^n$ is a well-defined bounded ...
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1answer
28 views

Linear map, inner products and orthonormal bases

If $L: \mathbb{V} \to \mathbb{V}$ is any map and we have two inner products defined on $\mathbb{V}$, $[ , ]$ and $⟨ , ⟩$, and we pick two orthonormal bases w.r.t. each of these inner products, how can ...
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Compute inner product $\langle (1+i,4) , (2-3i, 4+5i) \rangle$ in $\mathbb{C}^2$

Compute inner product $\langle (1+i,4) , (2-3i, 4+5i) \rangle$ in $\mathbb{C}^2$ My Answer: $= (1+i)(2-3i) + 4(4+5i)$ Book's answer: $=(1+i)(2+3i)+4(4-5i)$ Why does the book's method ...
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Why conjugate when switching order of inner product?

There is an axiom of inner product spaces that states: $\overline{\langle x,y\rangle } = \langle y,x\rangle$ Basically (without any conceptual understanding) it seems like all you have to do when ...
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Subsec. 4.1-8 in Kreyszig's functional analysis book: Does every inner product have a total orthonormal set?

In every Hilbert space $H \neq \{0 \}$, there exists a total orthonormal set. I think I've understood the proof given by Erwin Kreyszig in Introductory Functional Analysis With Applications. ...
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1answer
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Inner Product: prove that $\langle w, v+v' \rangle = \langle w, v \rangle + \langle w, v' \rangle$

We could use linearity in the first argument, homogeneity in first argument, and conjugate symmetry properties of the dot product. So this was my attempt at proving this: We know that $\langle ...
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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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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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Given a vector space with two inner products, there is a linear transformation taking one to another

I am looking for some hint to the following question: Let $V$ be an $n$-dimensional real inner product space and let $\langle x,y\rangle$ and $[x,y] $ both be two different inner products on V. ...
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1answer
33 views

Find a function $h$ such that $g(x) =\langle f, h \rangle$

Let $P_2(\mathbb R)$ be an inner product space with $\langle f, h \rangle = \int_{0}^{1}f(t)h(t)dt$. Let $g(f) = f(0) + f'(1)$. Find $h(t)$ such that $g(f) = \langle f,h \rangle$. I tried ...
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Prob. 9, Sec. 3.9 in Erwin Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS: Finite-dimensional range and the form of images

Let $H$ be a Hilbert space, and let $T \colon H \to H$ be a bounded linear operator. Then how to show the following? The range of $T$ is finite-dimensional if and only if $T$ can be represented in ...
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2answers
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Is a subset of an inner product space also an inner product space?

My question may seem trivial but it's important that I know this. I know for a fact that a subspace of an inner product space is also an inner product space, but how about an arbitrary subset? Could I ...
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1answer
48 views

Why is $V^{\perp}={0}$

Let $V$ be an inner product space. I have read a statement saying $V^{\perp}=\{0\}$. Why is this true? It seems trivial to even define an orthogonal complement to $V^{\perp}$ if it is always just $0$. ...
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Prob. 4, Sec. 3.9 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS: Image of a set under the adjoint operator

Here's Prob. 4, Sec. 3.9 in Introductory Functional Analysis With Applications by Erwine Kreyszig: Let $H_1$ and $H_2$ be Hilbert spaces, and let $T \colon H_1 \to H_2$ be a bounded linear ...
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Proof of basic Inner Product Property

I am searching for a proof of the following property for two vectors x and y in $\mathbb R^n$ $<x,y> = ||x||*||y|| \implies ||x||= \lambda*||y||$ for some $\lambda\in\mathbb R$ with ...