2
votes
3answers
41 views

The “Circle” is a Vector Space?

Consider the set of angles $C = [0, \ 2\pi)$ and, for all $x,y \in C$, define the $sum$ operation as the sum modulo $[0, \ 2\pi)$. The identity element of the addition is the angle $0$. The inverse ...
2
votes
1answer
52 views

Series in a space which is not complete

Let $X$ be a normed vector space and $\left\lbrace x_n \right\rbrace_{n \in \mathbb{N}} \in X^{\mathbb{N}}$ with $$\sum_{n=1}^{\infty} \|x_n\| < \infty \wedge \sum_{n=1}^{\infty} x_n \notin X,$$ ...
0
votes
0answers
14 views

semi linear uniform space

In semi-linear uniform space, if $f$ is a function from $(X ,Γ_X)$ to $(Y,Γ_Y)$ that is linear and bounded ,is $f$ then continuous? Is the converse true?
0
votes
1answer
20 views

fields and subspaces

Let F be a field and let V=F^F, which is a vector space over F. Let w be the set of all functions f element of V satisfying f(1)=f(-1). Is W a subspace of V? a. Has the zero vector b. closed under ...
0
votes
1answer
60 views

Show that Z cannot be turned into a vector space over any field. [duplicate]

Show that Z cannot be turned into a vector space over any field. So, we have 2 cases here. Case 1:lets suppose the charF=P, n does not equal 0, then (1+1+...+1)n=1n+1n+...+1n=n+n+...+n=pn=wchich ...
1
vote
0answers
25 views

Show that a function $\phi:(V, \|\cdot\|_{C^1}) \rightarrow (W, \|\cdot\|_{\infty})$, $f \rightarrow f'$ is continuous

I have the following: Show that the function $$\phi:(V, \|\cdot\|_{C^1}) \rightarrow (W, \|\cdot\|_{\infty}),~~f \rightarrow f'$$ is continuous. With $$\|\cdot\|_\infty = \sup\{|f(x)| ~ \big| ~x\in ...
3
votes
0answers
48 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 ...
2
votes
1answer
61 views

Is there a metric space and meanwhile a linear space such that vector addition discontinuous but scalar multiplication operation continuous?

Some special problems about topological groups or topological linear space theory. Recently I have done some study in some respects about topological group, topological linear spaces. And I found it's ...
0
votes
1answer
52 views

Proving Density of Subset of Hilbert Space

Suppose we have a subspace, $M$, of Hilbert space $H$. Prove the first statement implies the second statement: 1) If $<f,g> = 0$ for any $g\in M$, then $f=0$ in $H$. 2) $M$ is dense in $H$. I ...
1
vote
1answer
70 views

Is there any non-translation invariant but homogeneous metric linear space?

A metric linear space is a metric space and vector space, and linear operation is continuous regarding to the metric. I know that a homogeneous, translation invariant metric $d$ can be used to define ...
1
vote
0answers
14 views

References for vector valued spaces like $\mathcal C^\infty(A,C_0^\infty(B))$

I'm reading a text that use the space $\mathcal C^\infty(A,C_0^\infty(B))$, where $A\subset \mathbb R^n$ and $B\subset\mathbb R^m$ are opens. I've found, in Topological Vector Spaces, Distributions ...
0
votes
1answer
101 views

$l_0$ is all sequences with finitely many non-zero terms. Show $W^\perp=\{y: <x,y>=0, x\in W\}=\{0\}$ where $W = \{x : <x,a>=0\}$.

Consider the inner product space $l_0$ consisting of all infinite sequences of complex numbers with only finitely many non-zero terms, with the inner product of $l^2$ (space of square summable ...
1
vote
1answer
27 views

Can we construct two sets and functions for the given conditions?

Can we construct two sets $A$, $B$ and two invertible functions (one to one) $f_A \in \mathbb{R}^n$, $f_B\in \mathbb{R}^n$ such that the following conditions are satisfied? The conditions are ...
2
votes
1answer
132 views

Could we write Fourier transform as a matrix?

I have heard that Fourier transform is a linear transformation. I have also heard that any linear transformation can be written as a matrix multiplication. (probably I'm missing some details in the ...
-1
votes
1answer
76 views

proving boundedness of a linear functional

Let $f$ be a linear functional on a normed vector space V and $f^{-1}(\{0\})$ is closed. Prove that $f$ is bounded
0
votes
1answer
271 views

closed subspace of normed vector space

Is every finite dimensional subspace of a normed vector space closed? If yes, please prove it or else give a counter example.
0
votes
0answers
133 views

Has this Principal Component Analysis (PCA) been done correctly?

I have a set of 3D data points, indicated by the blue color in the picture below. I then project them onto the x-y plane, i.e. setting z values of all the points to 0, shown by the yellow color ...
0
votes
1answer
38 views

Show that $span({v_1,…v_n})=span({v_1,…,v_{n-1},w})$

This is what's given: $v_1,...v_n,w ∈ V$ and $v_1+...+v_n+w=0$ then I need to show that $span({v_1,...v_n})=span({v_1,...,v_{n-1},w})$ I could think of a way to show that this is true if I was sure ...
0
votes
2answers
66 views

Which of these sets is a subspace of F?

Let $F = \mathbb{R}^\mathbb{N}$. I need to check which of these sets are subspaces of $F$: $F_1 := \{ x \in F:\ \text{$x$ is bounded}\}$, $F_2 := \{ x \in F:\ \text{$x$ is convergent}\}$, $F_3 := \{ ...
3
votes
3answers
116 views

How to prove that sequence spaces $l^{p}, l^{\infty}$ and function space $C[a. b]$ are of infinite dimension

I am studying about the sequence space $l^{p}, l^{\infty}$ and function space $C[a. b]$. It is mentioned in the book that all of these spaces are of infinite dimension. I want to prove that these ...
0
votes
1answer
66 views

Pointed Convex cone: one-to-one correspondence extreme rays - extreme points

Hoi, let $V$ be a finite dimensional real vector space with inner product $\left\langle .\right\rangle$. Let $\Gamma\subset V$ and $\Gamma \neq \left\{0\right\}$ a pointed convex cone. (Pointed means ...
2
votes
1answer
46 views

On the existence of a bounded linear functional

Let $\mathcal{H}$ be a Hilbert space. By the Riesz Representation Theorem, we have that any bounded linear $\psi \in \mathcal{H}^{*}$ is of the form $\psi(h) = \langle h, g \rangle$ for some $g \in ...
0
votes
1answer
90 views

Distance of point for a set in linear spaces

Let $X$ a normed linear spaces, $Y \subset X$ a subspace and $z \in X$ an arbitrary point. How can we show that: $$\text{dist} (z, Y) = \sup \{\psi(z) \ | \ \|\psi\| = 1, \psi \equiv 0 \ \text{on} \ ...
3
votes
3answers
224 views

Can a non-zero vector have zero image under every linear functional?

Let $X$ be an infinite-dimensional vector space, and let $x_0$ be an element of $X$ such that $f(x_0)=0$ for every linear functional $f$ defined on $X$. Then can we prove that $x_0$ is the zero vector ...
3
votes
2answers
276 views

How to prove that two non-zero linear functionals defined on the same vector space and having the same null-space are proportional?

Let $f$ and $g$ be two non-zero linear functionals defined on a vector space $X$ such that the null-space of $f$ is equal to that of $g$. How to prove that $f$ and $g$ are proportional (i.e. one is a ...
2
votes
1answer
124 views

Equivalent statements of continuity of linear operators

I am asked to prove that the following are true: Given a linear operator $T: X \to Y$ where $X,Y$ normed linear spaces: (1) $T$ continuous at at point $\iff$ $T$ continuous everywhere (2) $T$ ...
1
vote
1answer
85 views

Sequential continuity in normed linear spaces

I am trying to prove the following "contiuity-type" result. Let $X,Y$ normed linear spaces. Let $\{T_n\} \to T \in \mathcal{L}(X,Y)$ and $\{u_n\} \to u \in X$. Show that $\{T_n(u_n)\} \to \{T(u)\} ...
4
votes
1answer
325 views

Image of unit ball dense under continuous map between banach spaces

I am assuming that the following problem will require the open mapping theorem, or maybe the closed graph theorem. Any help that can be given will be deeply appreciated. The statement is the ...
4
votes
2answers
117 views

Normed linear space and linear functional

Let $X$ be the normed linear spaceof sequences of reals that have only finitely many non-zero terms. Given $x = \{x_n\} \in X$, define $$f(x) = \displaystyle \sum_{n=1}^{\infty} x_n$$ I think that it ...
4
votes
1answer
233 views

Looking for proof that an open set in vector space contains the sum of two open sets.

Problem: To show that, in a topological vector space, for a given neighborhood of zero $W$, there exist two neighborhoods of zero, $V_1$, $V_2$, whose sum is contained in the first neighborhood, ...
0
votes
0answers
115 views

Prove that if $L(S)$ is bounded, where $S$ is the unit sphere of $U$, then $L$ is Lipschitz.

Let $U$ and $V$ be normed linear spaces over $\mathbb{R}$, and $L : U \mapsto V$ a linear function. Prove that if $L(S)$ is bounded, where $S$ is the unit sphere of $U$, then $L$ is Lipschitz. There ...
0
votes
1answer
25 views

Dimension of coefficents in a density equation

The density throughout a composite material is given by $T(x, y, z) = Axy^2 + Bxz^3 + Cy^2z^3,$ where $x$, $y$ and $z$ are the cartesian coordinates of the position inside the material. (a) Find the ...
0
votes
1answer
58 views

How to find the derivative with respect to the transformed co-ordinates.

I am stuck with something very simple , would be glad to get help . Suppose if i have a transformation matrix J , how do i find the derivative with respect to new co-ordinates , and derivative of ...
0
votes
2answers
69 views

Intersection of $|z_1 - x|=r$ and $|z_2 - y|=r$

Let $x,y \in \mathbb{R}^k$ ($k\geq 3$), $|x-y|=d>0$ and $r>0$. Prove that if $2r>d$, then there are infinitely many $z\in \mathbb{R}^k$ such that $|z-x|=|z-y|=r$. Here's what I have ...
1
vote
1answer
132 views

Prove: If $L \leq X$, $L$ has finite dimension, $M\leq X$ Then $L+M$ is closed.

Prove: If $X$ is a locally convex space, $L \leq X$, $L$ has finite dimension, $M\leq X$ Then $L+M$ is closed. What I know: If $L$ is a finite dimensional subspace, then $L$ is closed.
1
vote
1answer
634 views

Annihilators and dimensions

This question is related to this question. I have learnt from the question in the link that the equality below does not always hold. So here is my new question: What is a necessary and sufficient ...
1
vote
2answers
27 views

Definition clarifications on functions of vectors and their derivatives

Definition clarifications would be appreciated: How do I interpret the following ? For $f: R^n\to R^m$, $Df(\vec\xi)(\vec{x})$ in differentiation of a vector function. I know it as a function that ...
2
votes
1answer
170 views

Analogue of the dot product for functions

I have been reading some articles and I see that there is an analogue of the dot product for functions in the form of an integral. However, I am confused by the fact that there seems to be 2 forms: ...
7
votes
1answer
505 views

Does convexity of a 'norm' imply the triangle inequality?

Given a vector space $V$ (for convenience, defined over $\mathbb{r}$), we call $d:V\rightarrow\mathbb{R}$ a norm for $V$ if $\forall \mathbf{u}, \mathbf{v} \in V$ and $\forall r \in \mathbb{R}$ we ...
8
votes
6answers
883 views

Showing $1,e^{x}$ and $\sin{x}$ are linearly independent in $\mathcal{C}[0,1]$

How do i show that $f_{1}(x)=1$, $f_{2}(x)=e^{x}$ and $f_{3}(x)=\sin{x}$ are linearly independent, as elements of the vector space, of continuous functions $\mathcal{C}[0,1]$. So for showing these ...