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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5
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112 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 V\\ ...
1
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2answers
30 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 ...
0
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1answer
19 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 ...
2
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1answer
41 views

Is the inverse of a bijective connectedness preserving map , on a complete real inner product space , also connected 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
32 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}) = ...
4
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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
17 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*} ...
1
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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
48 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
25 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 ...
1
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1answer
27 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 ...
1
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2answers
29 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 ...
2
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2answers
48 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
19 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 ...
0
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2answers
43 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} = ...
1
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1answer
25 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 ...
3
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2answers
45 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. ...
1
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1answer
29 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, ...
0
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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
38 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 ...
1
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1answer
15 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 ...
0
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1answer
39 views

How to do this last step in this proof that inner product preserving implies linear?

Let $\tau : \mathbb R^m \to \mathbb R^m$ be a map such that $\tau (0) = 0$ and $\langle x,y \rangle = \langle \tau(x) , \tau(y) \rangle$ for all $x,y \in \mathbb R^m$. I want to show that $\tau$ is ...
1
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1answer
24 views

How can we derive the projection formula in general?

The derivation of the well-known projection formula $proj_\vec{b}(\vec{a})=\frac{\vec{a}\cdot \vec{b}}{\vec{b}\cdot \vec{b}}\vec{b}$ uses an argument based completely on geometry. We assume vectors ...
2
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2answers
29 views

A Proof of a False Result: If $U$ is $T$-invariant, then so is $U^\perp$.

$\newcommand{\ab}[1]{\langle #1\rangle}\DeclareMathOperator{\tr}{trace}\newcommand{\mc}{\mathcal}$ I have a "proof" of the following wrong fact: Let $T$ be a linear operator on a finite ...
0
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0answers
32 views

What is the minimizer of the matrix norm and it's significance?

For $M_{n\times n}$ a p.s.d real matrix, if we minimize $||M^{\frac{1}{2}}x||_2$ over $x$ under a linear constraint on $x$ as in $Ax=b$, where $b$ is non-zero. what is the significance of this $x$? ...
2
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0answers
37 views

Gradient of inner product in Hilbert space

Let $\mathcal{H}$ be a Hilbert space and \begin{align} f&\colon \mathcal{H} \to \mathbb{R}\\ f(x) &= ||x-c||_\mathcal{H} ^2 \end{align} from some constant $c \in \mathbb{H}$ Is the derivative ...
1
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1answer
33 views

Understanding of non-degeneracy and inner product

My class version of the non-degeneracy definition states: Let $V$ be a vector space over a field $\Bbb F$, equipped with a symmetric bilinear form $b : V \times V → \Bbb F$ . Then $b$ is a ...
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0answers
26 views

Is the subspace generated by complete othogonal subspaces closed?

$E$ is a vectorial space equipped with an inner product $\langle \cdot, \cdot \rangle$. $(E_i)_{i \in I}$ is a family of complete pairwise orthogonal subspaces. Is the subspace $V$ generated by the ...
2
votes
1answer
51 views

Proving that $T:\mathbb R^N \rightarrow \mathbb R^N$ is not surjective

Let $T:\Bbb R^n \rightarrow \Bbb R^n$ be a linear transformation and let $u_1,u_2$ different vectors in $R^n$ such that for every $v \in \Bbb R^n$ $Tu_1 \cdot v = Tu_2 \cdot v$. Prove that $T$ is ...
0
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1answer
43 views

Eigenvalues of Inner Product Matrix

Matrices have a least two major functions in linear algebra. On one hand, they can represent linear transformations as elements of $\text{Hom}(V,V)$). On the other hand they can represent inner ...
8
votes
3answers
382 views

Is there a name for this type of vector norm?

In the case of the $\mathcal{l}_2$ norm we have, $$||\mathbf{x}||_2^2=\mathbf{x}^T\mathbf{x}.$$ I was wondering if there was a type of norm that had a linear operation embedded in it, like this, ...
4
votes
2answers
125 views

What do we call the covector associated to a vector?

Let $V$ denote an inner product space. Write $V^*$ for either the algebraic dual, or else the continuous dual. In either case, for each vector $v \in V$, we get a covector $v^c \in V^*$ given by: ...
0
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1answer
42 views

Cauchy Schwarz inequality using L1 norm

1) Cauchy-Schwarz inequality states that the absolute value of vector inner product is always less or equal to product of norms of individual vectors i.e., $|a^Tb|\leq\Vert a\Vert_2 \Vert b\Vert_2$. ...
4
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0answers
26 views

If for every operator represented by $A$ w.r.t to a basis $\mathcal{B}$, the matrix representation of $T^*$ is $A^*$, then $\mathcal{B}$ is orthogonal

Let $V$ be a finite-dimensional inner product space. Assume that for every linear operator $T$, represented by $A$ w.r.t to a basis $\mathcal{B}$, the matrix representation of the adjoint w.r.t to ...
5
votes
7answers
719 views

Cauchy-Schwarz inequality proof (but not the usual one)

Before you downvote/vote-to-close, I am not asking for a proof of: $$\sum^n_{i=1}a_ib_i\le\sqrt{\sum_{i=1}^na_i^2}\sqrt{\sum^n_{i=1}b_i^2 } $$ Which is what EVERY link I've found assumes is the ...
0
votes
3answers
89 views

Find the function $\hat{g}$ that maximizes $\int_0^1 x^2g(x)dx$ over the set of all functions that satisfy the following conditions:

Find the function $\hat{g}$ that maximizes $\int_0^1 x^2g(x)dx$ over the set of all functions that satisfy the following conditions: $\int_0^1 |g(x)|^2dx =1$, $\int_0^1 g(x)dx=0$, and ...
4
votes
1answer
66 views

History of inner products and texts on it?

Where does the inner product originate from, was it defined in term of the dual or was it defined from just two copies of the space? I.e $(*,*) : V \times V \rightarrow scalar $ or $(*,*) : V \times ...
2
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1answer
28 views

Similarity of real symmetric matrices

I've thought about this question for about an hour but I'm still not able to arrive at correct answer. Can anyone suggest me how to go about it?
0
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1answer
26 views

Scalar triple product of quaternion scalar parts

I'm reading this paper about quaternion and 3D rotation with unit quaternions, \begin{eqnarray*} && \dot{q} = (q, {\bf q}) \\ && \dot{r} = (r, {\bf r}) \\ && \dot{r'} = ...
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0answers
23 views

Is the direct sum of two orthogonal subspaces well defined in infinite-dimensional vector spaces?

Let's say that $V$ is an inner product space on some field $\Bbb{K}$ and $M$ is a subspace of $V$. If $M^{\perp}$ is the orthogonal complement of $M$ with respect to the inner product, can I make the ...
7
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1answer
66 views

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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0answers
25 views

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 ...
5
votes
2answers
276 views

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 ...
3
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0answers
41 views

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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3answers
52 views

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: $p = \pi/3$ $p = \pi/4$ $p = \pi/2$ $p = 2\pi/3$
2
votes
1answer
80 views

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. ...
4
votes
1answer
46 views

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 ...
0
votes
0answers
14 views

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 ...
-1
votes
1answer
53 views

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 $A-mI\geq 0$. ...
0
votes
1answer
17 views

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 ...