3
votes
0answers
45 views

What extra assumption makes this transformation affine?

Let a vector space $V$ be given. Let $f:V\to V$ have the property that for all $x,y,a\in V$, $$ f(x+a)-f(y+a) = f(x) - f(y) \tag{$\star$} $$ Q1. I'd like to know how weak one can make additional ...
1
vote
1answer
21 views

why a lemma shows well-definedness of linear transformations

The following lemma can be used to show that some linear transformations are well-defined. I don't quite see that. I mean, if a linear transformation $T$ is well-defined, then if $x=y$ then ...
1
vote
2answers
58 views

What does this theorem mean?

Let $(V,\|\cdot\|)$ be a finite-dimensional normed space. Define $\|T\|_\mathrm{op}=\sup\{\|T(x)\|:\|x\|≦1\}$, for all linear operators on $V$ Define $\Omega$ to be the set of all invertible linear ...
1
vote
0answers
25 views

A System of Matrix Equations

I am looking for the solutions to the following system of matrix equations: (1) $AA^{*} - A^{*}A = C^{*}C - BB^{*}$ (2) $DD^{*} - D^{*}D = B^{*}B - CC^{*}$ (3) $AC^{*} - C^{*}D = A^{*}B - ...
0
votes
0answers
19 views

What's an application of that the set of invertible linear operators is open?

Let $V$ be a finite-dimensional normed space. Let $\Omega$ be the set of all invertible linear operators on $V$. Then, $\Omega$ is open in $(\mathscr{L}(V),||\cdot||_{op})$ and $f:\Omega\rightarrow ...
0
votes
1answer
33 views

What is *the domain* of the definition of the operator norm.

My definition: Let $(V,||\cdot||),(W,||\cdot||)$ be normed spaces over $\mathbb{F}$. Let $T:V\rightarrow W$ be a continuous linear transformation. Then $||T||_{op}\triangleq ...
0
votes
2answers
30 views

How do i prove that continuous linear transformation between normed spaces is bounded?

Let $(V,||\cdot||_V),(W,||\cdot||_W)$ be normed spaces over $\mathbb{C}$. Let $T:V\rightarrow W$ be a linear transformation. Assume $T$ is continuous. Then, how do i prove that $\exists c>0$ ...
1
vote
1answer
30 views

Inequality for norm of linear combination of linearly independent vectors

I'm trying to find a proof for the following: Let {$u_{1},...,u_{n}$} be a linearly independent set of a normed space $X$. Then, there is a constant $c>0$ such that for every set of scalars ...
0
votes
0answers
44 views

Triangle Inequality for matrix norm

$A$ and $B$ are square matrices, show that $σ_{max}(B) ≥ |σ_{min} (A+B) - σ_{min} (A)|$ I know that the induced 2-norm of of $B$ is $σ_{max}(B)$ but I don't know how to proceed with RHS.
2
votes
2answers
74 views

Can anyone explain this isometry to me? $T: (\mathbb R^n,\|\cdot\|_1)^*\to (\mathbb R^n,\|\cdot\|_{\infty})$, $ T(x)(y) = \sum_{i=1}^n x_i y_i$

Can anyone explain this isometry to me? $$T: (\mathbb R^n,\|\cdot\|_1)^*\to (\mathbb R^n,\|\cdot\|_{\infty}),\qquad T(x)(y) = \sum_{i=1}^n x_i y_i$$ I don't get what the domain and image of $T$ are. ...
1
vote
1answer
41 views

question about norms and convex set

Suppose $\overline{B}(0;1) = \{ x \in X : ||x|| \leq 1 \}$ is the closed unit ball on a vector space $X$. MY question is: is the following true? If $\overline{B}(0,1) $ is not convex, then $|| \cdot ...
1
vote
2answers
39 views

For inner product spaces, do we have $||\vec{u}-\vec{v}|| \leq ||\vec{u}||+||\vec{v}||$?

Let $V$ be an inner product space. Then for all $\vec{u},\vec{v} \in V$ we have $$||\vec{u}-\vec{v}|| \leq ||\vec{u}||+||\vec{v}||.$$ I know that the converse to the equation is true such that ...
0
votes
1answer
37 views

Continuous linear functional

I want to show that $f:(\ell^1,\parallel. \parallel_1)\to \mathbb K$ defined by $f((x_n))=\sum\limits_{n=1}^{\infty}\dfrac{\vert x_n\vert}{n}$ is continuous linear functional and the norm of $f$ is ...
1
vote
2answers
61 views

Inequality regarding norm vector space

I am not sure how to prove this inequality involving norms. Let $X$ be a normed vector space and $x,y$ are vectors in $X$ with nonzero norms. Prove the following inequality is true. $$\|x-y\|\geq ...
2
votes
1answer
57 views

The density of diagonalizable matrices of $M_n(\mathbb{C})$ problem.

For any matrix $A = (a_{ij})_{1\leq i,j\leq n} \in M_n(\mathbb{C})$, we pose $||A|| = \max_{1\leq i,j\leq n} |a_{ij}|$. $1.$ Show that $||.||$ define a norm on $M_n(\mathbb{C})$ and that $\forall A, ...
1
vote
0answers
66 views

Every inner product space is a normed space which is also a metric space

As I am pretty sure that everybody knows that a Hilbert space is a space that is a complete, separable and generally infinite dimensional inner product space. By the means of completion, every Cauchy ...
5
votes
1answer
67 views

Prove, that f is a linear map.

$U,V$ - Euclidean spaces $f:U \rightarrow V$ $f(0)=0$ $ \forall _{u,v \in U}:d(f(u),f(v))=d(u,v)$ Prove that $f$ is a linear map. I'm thinking about something like this: $||f(u+v)|| =d(f(u+v),0) = ...
3
votes
1answer
77 views

Is there always an injective map from a space in its dual space?

Today our teacher said that dual spaces are "big" and told us that this is a consequence by Hahn-Banach's theorem. So I was wondering whether the dual space of a space is always "bigger" or equal ...
0
votes
1answer
69 views

Show that orthogonal complement is trivial

I have this subspace of $C[-1,1]$ with inner product $\langle f,g\rangle = \int_{-1}^1f(x)\cdot \bar g(x)\,dx$: $$ E=\left\{f : \int_{-1}^0f=\int_{0}^1f\right\} $$ need to prove that $E^\bot=\{0\}$
0
votes
1answer
31 views

“Almost orthogonalizing” matrices using a signature matrix

Suppose $A$ and $B$ are two real symmetric $n \times n$ matrices (If simpler, consider $A$ and $B$ to be 0/1 matrices, say, adjacency matrices of d-regular graphs). Then $||AB||_{op} \leq ...
0
votes
2answers
41 views

If $||f-g|| < ||f^{-1}||^{-1}$, then $f$ is isomorphism implies $g$ is also isomorphism

Let $E$, $F$ be Banach space, $f,g \in L(E,F)$ and $f$ is isomorphism. Prove that if $||f-g|| < ||f^{-1}||^{-1}$, then $g$ is isomorphism. Hi everybody. I got stuck on this problem and can't ...
0
votes
0answers
217 views

When is an injective linear operator bounded below?

Given an injective linear operator $T$, densely-defined as a map between two normed linear spaces $V \rightarrow W$, what is a necessary and sufficient condition to enforce on $V$ to ensure the ...
2
votes
1answer
64 views

Infinite line is closed in $\mathbb{R}^n$

I have been reading the book "Elements of the functional analisys", by Kolmogorov and Fomin. At the chapter of Normed Linear Spaces, page 73 to be precise, the author makes the following definitions: ...
-3
votes
1answer
55 views

Conditional number: exercise

Let's say we've got a $202 \times 202$ matrix $A$ for which $||A||_2=100$ and $||A||_F=101$ (the Frobenius norm). How can we find the sharpest bound (lower) on the 2-norm condition number of $A$? ...
1
vote
0answers
48 views

Show that $S_x$ contains an interval

$X$ is Normed vector space, and $S_x={\{x\in X\space|\space\space\space||x||=1\}}$. Prove that if exist $x,y\in X$ which are linearly independent such that $||x+y||=||x||+||y||$, hence exist $u,v\in ...
0
votes
1answer
25 views

Norms on $\mathcal{M}_{n \times n}(\mathbb{R})$

On the space of all matrices $n \times n$ with real coefficients $\mathcal{M}_{n \times n}(\mathbb{R})$ we define two norms: $||A||_1 := sup_{x\neq 0; x \in \mathbb{R}^n} \frac{||Ax||}{||x||} \\ ...
9
votes
1answer
176 views

Proof that multiplying by the scalar 1 does not change the vector in a normed vector space.

I'm beginning a self-study of functional analysis, and I seem to have come to a halt trying to solve the first problem in the first problem set, and was wondering if someone could give me a pointer. ...
2
votes
1answer
346 views

Difference between convergence in norm, point-wise and uniform convergence

I know both definitions but I was wondering what are the relations between them. My question is if someone could explain intuitively the differences between these types of convergence. Specifically, ...
2
votes
1answer
80 views

Projection on unit vector lipschitz condition

if i have a vector $ v \in C^{n}$, where we do not want to look at vectors with norm $||v||\le 1$ with an arbitrary norm of this space and I am asked whether the map $ f(x)=\frac{x}{||x||}$ satisfies ...
8
votes
2answers
140 views

Two terms that I want to understand: weakest topology and jointly continuous (in the following context).

I was reading an article online, please help me to understand the following lines (in bold letters). - Topological structure: If (V, ‖·‖) is a normed vector space, the norm ‖·‖ induces a metric and ...
0
votes
1answer
164 views

How to expand equation inside the L2-norm?

I want expand an L2-norm with some matrix operation inside. Assume I have a regression $Y=X\beta+\epsilon$. I want to solve (meaning expand), $$\displaystyle\|Y-X\beta \|_{2}^2$$ Should I do: 1) ...
1
vote
1answer
165 views

Relation between Normed space and inner product

Below is what i have proved: If $V$ is a normed vector space over $\mathbb{R}$ satisfies parallelogram equality, then there exists an inner product $\langle \bullet,\bullet\rangle$ such that ...
3
votes
3answers
352 views

difference between normed linear space and inner product space

I've seen that the definitions of normed linear space and inner product space for a complex vector space $V$ are very close to each other except for the fact that one is defined on $V$ and the other ...
2
votes
2answers
105 views

Picard iterations of a matrix

I need help with this problem. I think i got the first three questions of the exercise, but i'm stuck at the fourth one. We consider the map $T:{\mathbb{R}^2}\longrightarrow{\mathbb{R}^2}$ defined ...
3
votes
1answer
116 views

If $U$ is finite dimensional, then operator norm is finite

Let $M:U\to V$ be a linear map between normed vector space $U$ and $V$. We know $U$ is finite dimensional (but don't know about $V$). Define $\|M\| = \sup \{\|Mv\|\;:\;\|v\| = 1\}$. I want to show ...
3
votes
1answer
54 views

How to show $A=\{T\in \mathcal{L}(\mathbb R^m, \mathbb R^n); \textrm{T is onto}\}$ is open in $\mathcal{L}(\mathbb R^m, \mathbb R^n)$

How can we show $A=\{T\in \mathcal{L}(\mathbb R^m, \mathbb R^n); \textrm{T is onto}\}$ is open in $\mathcal{L}(\mathbb R^m, \mathbb R^n)$? Here $\mathcal{L}(\mathbb R^m, \mathbb R^n)$ is the set of ...
14
votes
1answer
481 views

Multiplicative norm on $\mathbb{R}[X]$.

How to prove that : there is no function $N\colon \mathbb{R}[X] \rightarrow \mathbb{R}$, such that : $N$ is a norm of $\mathbb{R}$-vector space and $N(PQ)=N(P)N(Q)$ for all $P,Q \in \mathbb{R}[X]$. ...
1
vote
1answer
285 views

Distance between real finite dimensional linear subspaces

Is there a usual distance between linear subspaces ($V,W$) of an n-dimensional normed vector space with inner product? In the case of hyper-planes one could use the angle (based on the inner product ...
1
vote
1answer
120 views

Reproducing Kernel Hilbert Space- notation and basics

Am reading about Reproducing Kernel Hilbert Space(RKHS) while reading through Functional Analysis and Hilbert Space material and am unable to get the notation : $k(·,xi)$ correctly. What does the dot ...
3
votes
1answer
125 views

The trace norm cannot be increased by composing with a unitary operator

$\newcommand{\tr}{\operatorname{tr}}$ I was reading a proof for the statement $|\tr(US)|\leq |\tr(S)|$, for every endomorphism $S$ on a complex vector space $H$ and every unitary operator $U$ on the ...
1
vote
1answer
38 views

How to show growth without bound in only certain cases and not in others?

I encountered the following problem. (We're working in a finite-dimensional real vector space, here.) Suppose $$A=\frac{1}{2}\left(\begin{array}{cc}-2 & 4\\1 & 1\end{array}\right).$$ Find ...
2
votes
1answer
72 views

Norms on inner product space over $\mathbb{R}$

Definition of the problem Let $\left(E,\left\langle \cdot,\cdot\right\rangle \right)$ be an inner product space over $\mathbb{R}$. Prove that for all $x,y\in E$ we have $$ \left(\left\Vert ...
8
votes
1answer
103 views

Inner product space over $\mathbb{R}$

Definition of the problem I have to prove the following statement: Let $\left(E,\left\langle \cdot,\cdot\right\rangle \right)$ be an inner product space over $\mathbb{R}$. prove that for all $x,y\in ...
0
votes
0answers
93 views

distance of an affine subspace to a polytope

I wonder how to prove the following statement. Let $V$ be a $d$-dimensional normed space with $d \geq 3$, let $P \subset V$ be a $(d-2)$-dimensional polytope. Then there is an $\epsilon > 0$ such ...
2
votes
1answer
570 views

Every norm on $\mathbb{R}^n$ is equivalent to $\|\cdot \|_\infty$?

We have just come across the lemma that every norm on $\mathbb{R}^n$ is equivalent to $\|\cdot \|_\infty$. My question is that do all norms on $\mathbb{R}^n$ take the form $\|\cdot \|_1, \|\cdot \|_2, ...
2
votes
1answer
444 views

A question on linear transformation

Let $T$ be a linear transformation on $\mathbb{R}^{4}$ whose standard matrix is $$\left(\begin{array}{rrrr} 1 & -1 & -1 & -1\\ 1 & 1 & 1 & -1\\ 1 & 1 & -1 & 1\\ ...
2
votes
1answer
202 views

The sup-norm of a diagonalizable operator

I want to get familiar with computing sup-norms of diagonalizable operators on $\mathbf{R}^n$. Suppose that I have a diagonalizable linear map $T:\mathbf{R}^n\to \mathbf{R}^n$ and I consider ...
1
vote
1answer
108 views

tensorisation of linear map

Let $X$ be a Banach space and $T \colon \ell^2\rightarrow \ell^2$ be a bounded linear map. Suppose that the linear map $T\otimes Id_ {X}:\ell^2\otimes X\rightarrow \ell^2\otimes X$ which maps $e_i ...
2
votes
2answers
367 views

Point-wise and Norm Convergence of Vectors in a finite dimensional space

I'm trying to prove the theorem, that states, that if I have a normed vector space with a finite dimension (so that each vector I can express as a linear combination $$ ...