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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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 ...
2
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2answers
17 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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0answers
11 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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18 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
24 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 ...
0
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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 ...
0
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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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0answers
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\\ ...
3
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2answers
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 ...
2
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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 ...
0
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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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27 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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3answers
88 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
17 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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17 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 ...
2
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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 ...
0
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1answer
36 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
139 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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2answers
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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+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 ...
2
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1answer
33 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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2answers
35 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 ...
2
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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 ...
4
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2answers
53 views

Show that $T$ is normal

Let inner product space $V$ (finite) above $\mathbb{C}$. Let the operator $T:V\to V$ s.t. $$T^2 = \frac{1}{2}(T+T^*)$$ Prove that $T$ is normal $(T^*T = TT^*)$ $T^2 - T = 0$ So I've tried the ...
0
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1answer
29 views

The spectrum of a polynomial of an operator, question about proof, why are the operators invertible?

I have a question about a proof. In the proof $\sigma(T)$ is $\{\lambda \in\mathbb{C}: T-\lambda I\text{ is not invertible}\}$. In the proof they use this lemma: Here is the proof, my problem is ...
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1answer
34 views

Proving space of skew-symmetric matrices is orthogonal complement of symmetric matrices

Problem: Prove that $\left\{ A \in \mathbb{R}^{n \times n} \mid A \text{ is symmetric}\right\}^{\bot} = \left\{ A \in \mathbb{R}^{n \times n} \mid A \ \text{is skew-symmetric}\right\}$ with $\langle ...
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2answers
33 views

Finding orthonormal basis for a subspace $W$ of the Euclidean space $\mathbb{R}^3$.

Problem: Let $\mathbb{R}^3$ be an Euclidean space. Find an orthonormal basis for the subspace $W$ defined as $x + 2y-z = 0$. Attempt at solution: So this is a plane in $\mathbb{R}^3$, so I guess I ...
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2answers
49 views

Prove there's a unitary linear operator

Let $u, v\in V$, where $V$ is a finite dimensional vector-space, such that $\|u\|=\|v\|$. Prove there's a unitary linear operator such that $T(u) = v$ So if there's such unitary linear operator, it ...
2
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1answer
20 views

if $E,F$, two bases are orthonormal then $T$ is unitary.

Let $T:V\to V$ and two bases of $V$: $E = \{v_1, \ldots, v_n \}$ and $F = \{T(v_1), \ldots, T(v_n)\}$. Prove: $E,F$ are orthonormal implies $T$ is unitary. So basically we want to prove ...
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2answers
57 views

Normalizing a basis

Let the basis $B = \{1,x,x^2\}$ which is orthogonal. Now, I've seen the following: $$\|1\| = \sqrt {\langle 1,1\rangle} = \sqrt {4\cdot 1\cdot 1} = 2 $$ $$\|x\| = \sqrt {\langle x,x\rangle} = ...
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1answer
26 views

Where did I go wrong with the Gram-Schmidt orthogonalisation process?

Problem: Let $\alpha = \left\{(1,2,0), (1,0,1), (2,3,1)\right\}$ be a basis vor $\mathbb{R}^3$. Apply the Gram-Schmidt orthogonalisation process to turn $\alpha$ into an orthonormal basis for ...
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2answers
46 views

Prove that the linear transformations are the same.

I have this lemma: If X is a complex inner product space and $S,T \in B(X)$ are such that $(Sz,z)=(Tz,z)\forall z \in X$, then $S=T$. $B(x)$ is the set of bounded linear operators from X to X. ...
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1answer
30 views

Prob. 3, Sec. 3.2 in Kreyszig's Functional Analysis Book: Is the space of all polynomials of a fixed degree complete? [duplicate]

Let $n$ be a given natural number, and let $X$ denote the vector space consisting of the zero polynomial and of all polynomials of degree at most $n$, with real or complex numbers as co-efficients, ...
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1answer
41 views

$L^2$ inner product split over sub-domains in $\mathbb{R}^3$

I have a bounded Lipschitz domains $\Omega, \Omega_1, \Omega_2 \subset \mathbb{R}^3$ such that $\overline{\Omega}=\overline{\Omega}_1 \cup \overline{\Omega}_2$ and $\Omega_1 \cap \Omega_2=\emptyset$. ...
3
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1answer
40 views

How is the orthogonal projection on to the span of the columns of a matrix determined by a chosen inner product?

I know that of course a orthogonal projection must be orthogonal for a chosen inner product. But how can I find a new orthogonal projection based on $P=A(A^TA)^{-1}A^T$, if I have dot product defined ...
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1answer
17 views

An inner product space has a vector related to an arbitrary collection of scalars

Let $V$ be a finite dimensional inner product space with inner product $\left\langle , \right\rangle$ and let $\beta = \{v_1,...,v_n\}$ be a basis for $V$. If $c_1,...,c_n$ are arbitrary scalars, show ...