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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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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24 views

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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1answer
37 views

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 ...
8
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1answer
87 views

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 - its actually a basis transformation right ? $$ F(y) = \int K(x,y) f(x) \text{d}x $$ ...
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1answer
20 views

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. ...
2
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1answer
27 views

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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33 views
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23 views

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 ...
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3answers
41 views

What is the connection between the two ways in which functions get represented as vectors when discussing inner product?

When I interpret the inner product and orthogonality of two functions, I consider it the following way. Say you have a basis for a function space $\{\sin(x), \cos(x)\}$ and you have two vectors $u = ...
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1answer
50 views

The choice of scalar factors in the proof of the Schwarz inequality

In this proof for the Schwarz Inequality, they seemingly arbitrarily choose $r = w\cdot w$ and $s =-(v\cdot w)$. Why did they make these selections? I don't understand where these values for $r$ and ...
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1answer
32 views

Euclidean geometry and $L_2(\lambda)$ space

Suppose $f,g\in L_2(I,\lambda)$ with $\lambda$ any probability measure and the norm $\| x\|=\sqrt{\langle x, x\rangle}$. Could we have the same geometric properties in this space as in the Euclidean ...
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2answers
35 views

Does two functions have to have an inner product of zero for just one interval to be orthogonal?

If I have a function space defined by the basis {$sin(x), cos(x)$}. And you have the vectors $v = (2,2)$ and $u = (-1,1)$. So $v$ reperesent a point, on the plane define by the basis, the function ...
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1answer
48 views

What is the geometric meaning of the inner product of two functions?

When it comes to inner product I have thus far only dealt with vectors, and so the concept is very intuitive because one can easily visualize two vectors and how they get multiplied, and it is clear ...
3
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33 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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1answer
7 views

find the value $(g-Pg)(x),x\in [0,1]$

Let $V$ be a closed subspace of $L^2[0,1]$ and let $f,g \in L^2[0,1]$ be given by $f(x)=x$ and $g(x)=x^2$. If $V^\perp =$span $ \{f\}$ and $Pg$ is the orthogonal projection of $g$ on $V$ then find ...
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12 views

Clarification on Sequence space

I have a trivial question but which I'm feeling confused. Is the sequence space a finite collection of vectors whose components are infinite or am I misunderstanding the concept?
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2answers
24 views

Intuition of a norm vector space/ infinite dimensional vector space

I'm finding it terribly difficult to build an intuition of what a norm vector space and an infinite dimensional vector space is. There aren't any good notes online that builds the intuition-most ...
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2answers
19 views

Matrix tensor indices

Suppose I have an orthonormal basis $$B = \left \{ u_{i} \right \}_{i=1}^{\infty}$$ Then for a matrix $K$, do I represent it as $$K = \sum_{j,k=0}^{\infty}k_{jk}\left ( u_{i}\bigotimes u_{j} ...
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12 views

Inner Product of Square Matrices

Let K$^{n*n}$ & M$^{n*n}$ be two square matrices, and K$\cdot$M= \begin{matrix} t_{11} & \cdots & t_{1n} \\ \vdots & \ddots & \vdots \\ t_{n1} & \cdots ...
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14 views

Dot Product of Square Matrices & Inner Product

I need some help! Thank you in advance. Let K$^{n*n}$ & M$^{n*n}$ be two square matrices, and K$\cdot$M= \begin{matrix} t_{11} & \cdots & t_{1n} \\ \vdots & \ddots ...
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13 views

Difference between orthogonal projection onto a point and onto a vector.

A trivial question although I'd like some good answers. Are there any mathematical difference? My vector calculus is a bit rusty.
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2answers
21 views

Gram–Schmidt algorithm used for obtaining the orthogonal and orthonormal

Why are both the algorithm used for finding the orthogonal and orthonormal basis the same? I'm relying on a set of slides given by by lecturer (known to be sloppy!) and I want to confirm if it should ...
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2answers
32 views

Projection of vector $v$ on $u$ in terms of inner product

$$ \mathrm{proj}_{v}(u) = \frac{\left \langle v,u \right \rangle}{\left \langle v,v \right \rangle}v=\left \langle \hat{v,}u \right \rangle\hat{v} $$ I am unable to follow from the second to the last ...
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19 views

Using non-standard inner products for alternative notions of matrix product

It seems intuitive to think of billinear forms on finite dimensional vector spaces as coresponding to positive definite, symmetric or hermitian matrices. In this language, the standard inner product ...
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1answer
26 views

What exactly is pairwise orthogonal?

Suppose there exists a basis $$B = \left \{ v_{1},...,v_{n}\right \}$$ and basis $$B' = \left \{ v_{1}',...,v_{n}'\right \}$$ Then, if $$\left \langle B,B' \right \rangle=0$$ then B and B' are ...
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1answer
19 views

is this inner product positive-definite?

$$\left \langle u, v \right \rangle = pu_{1}v_{1}+qu_{1}v_{2}+qu_{2}v_{1}+pu_{2}v_{2}\\\text{ for }\\ \text{p >0} \text{ and } p^{2}\geq q^{2}$$ The solution breaks down $$\left \langle u, u ...
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2answers
23 views

bilinear/ linearity proof for the inner product property

One of the properties for inner product says: $$\left \langle \lambda u,v \right \rangle = \lambda\left \langle u,v \right \rangle$$ for all scalar lambda. $$\left \langle u_{1}+u_{2},v \right ...
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1answer
42 views

Bounds for inner product of $Ax$ and $x$

Reading a math text, I found, with no proof given, the following assertion. Suppose $A$ is a real $n \times n$ matrix, and suppose the real part of its spectrum lies between $a$ and $b$; i.e., the ...
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29 views

Solutions to the equation $\langle z,u_1\rangle \langle z,u_2\rangle =\langle u_1,u_2\rangle$.

Suppose $u_1$ and $u_2$ are elements of $\mathbb{C}^N$ of norm $1$, and that $\langle u_1,u_2\rangle\not =0$. If $z$ is in $\mathbb{B}_N$ (the unit ball of $\mathbb{C}^N$), how many solutions does ...
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9 views

On equality of frobenius norms with constraints

For a given matrix A, \begin{equation} \begin{aligned} &&&\| Z \|^2_F = \| A \|^2_F\\ &&& Tr(Z'Z) = Tr(A'A)\\ & \text{subject to} & & Z \succeq 0\\ ...
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36 views

Question on a property of the orthogonal complement: $A\subset(A^\perp)^\perp$

Among the principal properties of the orthogonal complement, we have the following: $$A\subset(A^\perp)^\perp$$ Where $A$ is a subset of an inner product space $X$, and $A^\perp$ is the orthogonal ...
2
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1answer
51 views

$\ell^p\!,$ for $p\neq2$, is not an inner product space. [duplicate]

Consider sequence spaces of the form, $$\ell^p=\Big\{x=\left(x_j\right)_{j=1}^\infty \mathrel{}\big|\mathrel{} \sum_{j=1}^\infty \left\lvert x_j \right\rvert ^p\lt\infty\Big\}$$ for $1\le ...
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2answers
43 views

Are there any books/papers talking about inner product on vectors over finite fields?

Are there any books/papers talking about inner product on vectors over finite fields? In particular, I'd like to learn things on $F_p^n$, or simply $F_2^n$. I read some proofs using the inner product ...
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2answers
79 views

prove $\langle f(x),f'(x)\rangle = 0$

Let $f: R \to R^n$ be a differentiable function such that $\forall x \quad||f(x)|| = 1$ prove that $\forall x \quad \langle f(x),f'(x)\rangle = 0$ i thought of the following proof but not sure it ...
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2answers
47 views

Possible ways to induce norm from inner product

Let $ S $ be a normed vector space (over $\mathbb{R}$, say, for simplicity) with norm $ \lVert\cdot\rVert$. Can this norm be induced from inner product only through $\lVert \cdot \rVert = ...
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1answer
33 views

Dual of Hilbert space : induced norm vs. operator norm

Let $\mathfrak{H}$ be a Hilbert space. Is the operator norm on the dual $\mathfrak{H}^*$ induced by a inner product ?
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89 views

Uniqueness of midpoints in inner product spaces

Does anyone have an elegant proof of the following fact? Let $V$ be a real inner product space and let $x$ and $y$ be two elements of $V$. If $z\in V$ is such that $\lVert x-z\rVert=\lVert ...
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17 views

Correspondence between Inner and Outer Products

For two $n\times p$ matrices $X$ and $Y$ (p-dimensional space), if its outer product is equal to zero i.e. $X^{T}Y = 0_{p}$, what can be said about its dual inner product matrix $YX^{T}$, or the vice ...
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1answer
19 views

Getting the unique element in the Riesz-Frechet Theorem.

I have this thorem in my book, H', denotes the dual space, that is the set of bounded linear operators from X to the field over X. The way they got the unique element seems very interesting. Does ...
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18 views

Find Isotropic vectors that form a basis

I have this question: let $(E,\langle,\rangle)$ an inner product space with dimension $n$ and $u$ a symmetric linear transformation and we define a quadratic form $q$ by $$\forall x\in E,\quad ...
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1answer
26 views

Finding rotated orthogonal vectors without knowing lengths

I have two abstract orthogonal vectors $\mid a\rangle$ and $\mid b\rangle$: $\langle a\mid b\rangle=0$, but I don't know the lengths $\mid a\mid=\sqrt{\langle a\mid a\rangle}$ and $\mid ...
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1answer
40 views

inner product and hermitian scalar product

suppose $\underline x,\underline y\in\mathbb C^{n\times 1}$ then because the two vectors are in complex vector field, the definition of their inner product will be: $$\langle\underline x,\underline ...
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1answer
140 views

Cauchy-Schwarz Inequality without using $\langle a x,y\rangle=a\langle x,y\rangle$

Let $V$ be a vector space and define a function $\langle .,.\rangle:V\times V\to\mathbb{C}$ such that $$\begin{align} & \langle x,y\rangle=\overline{\langle y,x\rangle }\,\,\,\forall x,y\in ...
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39 views

Orthonormality of the columns of a matrix

I am studying orthogonal columns and matrices right now and I have encountered the following theorem: Theorem An $m \times n$ matrix $U$ has orthonormal columns if and only if $U^T U = 1$. Is it ...
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1answer
24 views

A simple proof for angle inequality in inner product spaces

I am looking for a smiple proof for the following fact: Let $u,v,w$ be vectors in an inner product space $V$. Then it holds: $\theta (u, v)≤\theta(u, w) + \theta(w, v)$ (Of course if they are all in ...
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87 views

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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1answer
34 views

Is taking the real part required in vector orthogonality and projection?

In a real inner product space, two vectors are orthogonal if $\langle \mathbf{u}, \mathbf{v} \rangle = 0$. Similarly, $$\operatorname{proj}_{\mathbf{u}}(\mathbf{v}) = ...
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37 views

Parallelogram law in $L_1$ space

Exercise 5.5 from Capinski's and Kopp's book "Measure, Integral and Probability" asks to show that it is impossible to define an inner product on the space $L^1([0,1])$. In order to get this result we ...
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1answer
21 views

Determining dimension of a sum of subspaces in terms of a parameter

Problem: Consider the linear subspaces \begin{align*} U = \text{span} \left\{ (1,0,1,0), (1,a,0,a)\right\} \quad \text{and} \quad W = \text{span}\left\{(-1, a, a^2, 0), (0,1,0,-1)\right\} \end{align*} ...
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2answers
25 views

Determining all scalars $a \in \mathbb{R}$ for which a matrixrepresentation is orthogonal?

Problem: Let $a \in \mathbb{R}$ and \begin{align*} T: \mathbb{R}^{n \times n} \rightarrow \mathbb{R}^{n \times n}: A \mapsto aA. \end{align*} Determine all $a \in \mathbb{R}$ for which the matrix of ...