1
vote
0answers
10 views

Request for information about certain linear transformations of functions on subsets

Suppose I have an infinite set $U$ and let $M$ be the linear subspace of all real-valued functions $\nu$ on $2^U$ such that $\nu(\emptyset) = 0$. Here the sum of two such functions (and the product of ...
1
vote
0answers
54 views

Proving boundedness of a function .

Consider the function \begin{eqnarray} f(x_1,x_2,\cdots, x_n) = \frac{\sum_{i}^{n}a_ix_i}{\sum_{i}^{n}b_ix_i}, \end{eqnarray} over the set $S = \{x := (x_1,x_2,\cdots, x_n):-1 \leq x_i \leq 1,\; ...
3
votes
1answer
45 views

Is the set of all Taylor polynomials a vector space?

Let $V$ denote the set of all Taylor polynomials of degree $\leq n$ for a fixed natural number $n$ (including the zero polynomial), regraded as real-valued functions of a real variable. Then is $V$ a ...
0
votes
1answer
49 views

Proving boundedness of a function (part 1).

Consider the function \begin{eqnarray} f(x_1,x_2,\cdots, x_n) = \frac{\sum_{i}^{n}a_ix_i}{\sum_{i}^{n}b_ix_i}, \end{eqnarray} over the set $S = \{x := (x_1,x_2,\cdots, x_n):-1 \leq x_i \leq 1,\; ...
1
vote
0answers
26 views

Conjunctive Normal Form representation/ First Order Logic.

in my research problem, I need to represent three types of three types of relationships between the variables x,y as the following:: " y Cooperates with x" relationship: means if there is two ...
0
votes
0answers
21 views

Proving that integrator operator of a kernel satisfies a specific peroperty

I am trying to prove that a integrator operator of a kernel satisfy a specific property say $\phi$. By integrator operator for non-negative definite kernel $\mathcal{K}$ I mean $T_{\mathcal{K}}$ such ...
0
votes
0answers
122 views

Span of Dirac's delta distributions dense in Hilbert space of $L^2$ functions?

According to Wiki a set of elements of a Hilbert space(B) is a basis for that space if: Orthogonality: Every two different elements of $B$ are orthogonal: $⟨e_k,e_j⟩=0$ for all $k$, $j$ in $B$ with ...
1
vote
1answer
37 views

defenite integral involve bessel function

I have an integral which involves Bessel function as follows: $I=\int_{r=0}^a \int_{\theta=0}^{2\pi}(e^{-jkr\cos(\theta-\phi)}d\theta)rdr$ I have tried with $e^{-jkr\cos(\theta-\phi)}=\sum ...
0
votes
3answers
81 views

Complex Roots and calculations

roots of the equation $z^6 =1-\sqrt3 i $ are $$z_1,z_2,z_3,z_4,z_5,z_6 $$ calculate:$$|z_1|^3 +|z_2|^3+|z_3|^3+|z_4|^3+|z_5|^3+|z_6|^3$$ also calculate: $$z_1^6 +z_2^6+z_3^6+z_4^6+z_5^6+z_6^6$$ ...
2
votes
0answers
62 views

Prove that the norm of $E$ is generate by the inner product $\langle x,y \rangle =\frac{1}{4}\left(||x+y|^2-||x-y||^2\right)$ [duplicate]

Let $E$ a normed linear space such that: $$\|x+y\|^2+\|x-y\|^2=2\|x\|^2+2\|y\|^2 $$ Prove that the norm of $E$ is generate by the inner product $$\langle x,y \rangle ...
1
vote
1answer
22 views

About dual space and density

Let $Z\leq X$ be a dense subspace. Prove that their duals are equals. I need to complete my idea. I know that $Z^*\subset X^*$. By the way, if $f\in Z^*$, I think that I can use the density to extend ...
2
votes
0answers
61 views

which limit is true? [closed]

Let $p_n(x)=a_nx^2+b_nx+c_n$ be a sequence of quadratic polynomials where $a_n, b_n,c_n\in\mathbb R$, for all $n\geq 1$. Let $\lambda_0,\lambda_1,\lambda_2,$ be distinct real numbers such that ...
2
votes
0answers
50 views

How to find out the closed form of a function from its parametric form?

In general suppose that we have a parametric curve given by: $$ x = \phi(t) \\ y = \psi(t) $$ Then if $\phi^{-1}$ exists it is easy to get $y$ as a function of $x$ in closed form: $$ y = ...
2
votes
2answers
38 views

Matrices and the Dot Product

Prove that real $n \times n$ matrix $A$ satisfies $Ax \cdot Ay=x \cdot y$ for all $x, y \in \mathbb{R}^n$ if and only if $| Ax| = |x|$. $\textbf{My Attempt}$ Write $Ax \cdot Ay= A^2 ( x \cdot y)$. ...
1
vote
1answer
37 views

About linear transformation

Find: a) $T:l^{2}\rightarrow l^{2}$ linear and discontinuous. b) $T: l^{2}\rightarrow \mathbb{R}$ linear and discontinuous. Where $l^{2}$ is a $l^{p}$ space. Anyone have a canonical example? If ...
2
votes
0answers
122 views

Solving an infinite non autonomous system of differential equations.

For all $\lambda\in\mathbb{R}$, let $J(\lambda)$ be the infinite matrix where $(J(\lambda))_{nn}=\lambda$, $(J(\lambda))_{n,n+1}=1$ for all $n\in\mathbb{N}$, and all other entries are $0$. This matrix ...
0
votes
6answers
54 views

Why is one of the conditions of a vector space that if I add two vectors, the sum must be within the space? [closed]

I have been working with vector spaces for a while and I now take for granted what the vector space does. I feel like I dont really understand why multiplication and addition must be defined on a ...
4
votes
2answers
145 views

What is the intuition and motivation behind a norm on a space?

I have a question regarding norms. I am looking for an intuitive understanding of what norms do. From what I know so far, in any arbitrary space, if I have a norm, then it appears that it allows me to ...
3
votes
1answer
210 views

Numerical verification of solution.

I have the non-linear equation \begin{align} &\left( {x}^{2}-1 \right) \left( -\frac{1}{4}\left({\frac { \left( 4\,{x}^{3}+2\, ex \right) ^{2}}{ \left( {x}^{4}+e{x}^{2}+f \right) ...
3
votes
1answer
90 views

Eigenvalues of a symmetric matrix with Lagrange multipliers

Problem: Using Lagrange multipliers, prove that all symmetric matrices $A \in \mathbb{R}^{n \times n}$ have all real eigenvalues. Proof: Consider $f: \mathbb{R}^n \rightarrow \mathbb{R}$ defined by ...
2
votes
1answer
43 views

Perturbation of linear independence

Given a linearly independent and orthonormal set $\lbrace u_1,\ldots,u_n\rbrace \in \mathbb{R}^d$. Exist $0<\varepsilon<1$ such that $B(u_i,\varepsilon)\cap B(u_j,\varepsilon)= \emptyset$ for ...
0
votes
0answers
41 views

Convergence of Matrix Power Series

If $A$ is a square matrix with complex entries, then $\| A\|$ is defined as the sum of the absolute values of the entries of $A$. I have shown that this matrix norm is homogeneous, subadditive, and ...
1
vote
2answers
52 views

Why is there an “absolute value” and a norm in the Schwarz Inequality?

This really bothers me, and I'm not sure if it's just that I'm not understanding it correctly. For the moment, assume we are working in a vector space $V$ over $\mathbb{R}^n$. Let $x,y \in V$. We have ...
4
votes
1answer
76 views

Is there any proposition in real analysis or linear algebra that can only be proved by contradiction?

By "only be proved by contradiction", I mean either it's probable that this proposition can only be proved using contradiction, or that no one has ever came up with a direct proof. An undergraduate ...
2
votes
1answer
38 views

Operator norm of orthogonal projection

I was assigned the following homework problem: "Let $P:\mathcal{H} \to \mathcal{H}$ be bounded and linear. Assume it satisfies $P^2 = P$ and $P^\star = P$. Show $\|P\| \le 1$." This isn't too hard ...
0
votes
0answers
23 views

Cannot use alternative definition of “nowhere dense” to show space of real sequences with only a finite # of nonzero terms is NOT complete?

Suppose that I define my space $V$ to the the space of real sequences with only a finite number of nonzero terms. Then, I define $V_n = (a_1,a_2,\ldots,a_n,0,0,\ldots)$. Then, it is that $V$ has a ...
0
votes
0answers
18 views

Robust LP question using box uncertainty model

I am trying to solve this robust LP problem by writing it as a QP $$\min_x x^TSx : \mu \leq r^T x , Ax \leq b$$ Under Box uncertainty model: $$R = \{r : \| r - \hat{r}\|_\infty \leq \rho\}$$ Here ...
2
votes
1answer
53 views

Showing that the set of functions in $C(I)$ which are monotone on some nontrivial subinterval of $I$ is of first category in $C(I)$.

Let $I = [0,1]$ and let $C(I)$ be the metric space of continuous functions on $I$ with the $L^{\infty}$ norm. I am trying to show that the set of functions in $C(I)$ which are monotone on some ...
2
votes
1answer
260 views

How to make this polynomial the zero polynomial?(recursively)?

Given a fixed $\beta \in \mathbb{R}$, I want to find the $c_0,...,c_n$ for arbitrary $n \in \mathbb{N}$ such that the polynomial \begin{align}P_n(z):=z(1-z) ...
0
votes
1answer
36 views

Matrix Inequality: $A^\top B A \preccurlyeq B$

Consider $A, B \in \mathbb{R}^{n \times n}$, $A$ invertible, $B \succ 0$. Say if the following holds: $$ A^{- \top} B A^{-1} \preccurlyeq B \ \Longleftrightarrow \ B \preccurlyeq A^\top B A. $$ I do ...
1
vote
1answer
54 views

Prove vectorspace of bounded functions with supremum-norm is complete and no hilbert space

I have the following: Consider the real vectorspace with bounded functions $$V = \{f:[0,1]\rightarrow\mathbb{R} | \exists C > 0 : f([0,1])\subset[-C,C]\}$$ and the supremum-norm $$||f||_\infty ...
1
vote
1answer
21 views

Notions of tangent plane at function

For a differentiable function $f : \mathbb R \to \mathbb R$ the equation of the tangent plane at $x_0$ is $0 = f'(x_0) x - y$. But some functions not differentiable like $\sqrt x$ at $x_0 = 0$ still ...
1
vote
1answer
35 views

How to find Area and Perimeter

hey guys can you solve these 2 questions for me shown in the image below. I've done this but I think it's not correct could you guys just show me the solution with working. Thanks in Advan =) God ...
0
votes
2answers
69 views

Eigen values and Eigen vectors

Let A be a 4x4 matrix with real entries such that $ \ -1,1,2,-2 \ $ are its eigen values.If $B=A^4-5A^2+5I$ ,where $I$ denotes the 4x4 identity matrix ,then which of the following statements are ...
8
votes
4answers
878 views

Is the determinant differentiable?

I was wondering, given an $n\times n$ square matrix with $n^2$ many entries, the function $\det:\left(a_1,a_2,\ldots,a_{n^2}\right)\to \textbf{R}$ which gives the determinant where $a_{k}$'s are the ...
1
vote
0answers
33 views

Is this functional linear?

I know it's trivial, but is this functional not linear? $\phi:\mathbb{R}[X]\ni p \rightarrow p(0)p \in \mathbb{R}[X]$ $$\phi(p+q)=(p+q)(0)\cdot(p+q)=(p(0)+q(0))\cdot(p+q)\ne\phi(p)+\phi(q)$$
0
votes
1answer
54 views

Solve this problem?

Here I have 2 lists, A and B. I am trying to find the connection items of list A got with items of list B. I know: b) all the items of A a) the 1st item of B b) number range from 0-255 A - B 0 ...
4
votes
1answer
71 views

Vector spaces isomorphic, then dual spaces isomorphic

If we know that there is a (topological) isomorphism between two Banach spaces $X,Y$ called $\phi \in L(X,Y)$. Then the appropriate isomorphism between the dual spaces $X',Y'$ is given by $\phi' \in ...
1
vote
2answers
39 views

Linear operator exists then differentiable?

Let $E_{\text{open}} \subseteq \mathbb{R}^n$, and let $\vec{x_o} \in E$. Let $\vec{f}: E \rightarrow \mathbb{R}^m$. If there exists a linear operator $A: \mathbb{R}^n \rightarrow \mathbb{R}^m$. such ...
0
votes
1answer
30 views

Linear transformation from $R^2$ to $R^2$.

Let $\vec{f}: \mathbb{R}^2 \rightarrow \mathbb{R}^2$, where $\vec{f} (\vec{x}) = (x+y^2, x^3+5y)$ and $\vec{x} = (x,y) \in \mathbb{R}^2$. Let $\vec{h} = (h_1, h_2)$ and $\vec{a} = (1,1) \in ...
0
votes
1answer
71 views

show that $f(x,y) =2x^2 + 3y$ is differentiable at $(0,0)$ by finding a linear function T

Here's the question: Prove that $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ defined $f(x,y) = 2x^2 + 3y$ is differentiable at $\begin{bmatrix} 0\\0 \end{bmatrix}$ by producing a linear function T and ...
1
vote
0answers
53 views

Transfer vector space properties to dual space

I am curious about this here (Actually, I don't know if my assumptions are true or not) a) Let $X$ be a Banach space that is isomorphic to $Y$, then $X^*$ is also isomorphic to $Y^*$. I sketched a ...
1
vote
0answers
60 views

show that $f(x,y) =2x^2 + 3y$ is differentiable at $(0,0)$ by producing a linear function

Here's the question: Prove that $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ defined $f(x,y) = 2x^2 + 3y$ is differentiable at $\begin{bmatrix} 0\\0 \end{bmatrix}$ by producing a linear function T and ...
1
vote
2answers
42 views

showing $\inf \sigma (T) \leq \mu \leq \sup \sigma (T)$, where $\mu \in V(T)$

I am trying to prove the following: Let $H$ be a Hilbert space, and $T\in B(H)$ be a self-adjoint operator. Then for all $\mu \in V(T)$, $\inf \left\{\lambda: \lambda \in \sigma (T) \right\}\leq \mu ...
0
votes
2answers
50 views

Is this a basis for the dual space?

There is an example on Wikipedia that I don't understand and I'd appreciate some help. They define $\mathbb R^\infty$ to be the space of all sequences that are zero except for finitely many indexes. ...
1
vote
2answers
43 views

Dual space (Wikipedia)

I am struggling to understand something on Wikipedia: ''If $V$ consists of the space of geometrical vectors (arrows) in the plane, then the level curves of an element of $V^*$ form a family of ...
2
votes
1answer
27 views

Fast argument to see that the dual map of a projection is a projection

If $X$ is a Banach space and $U,V$ are closed subspaces, such that $X \cong U \oplus V$, then a continuous linear map $P:X \rightarrow X$ is called a projection if $P|_U =id$ and ker(P)=V. Now we ...
2
votes
0answers
97 views

A probability problem with multivariate Gaussian distribution

I am a computer science guy, not a mathematician so kindly excuse me if there is any ridiculous error in my problem description. I have two clusters $C_1$ and $C_2$ in a feature space spanned by $k$ ...
0
votes
1answer
29 views

Show that the norms $ | p | _1 $ and $ | p | _2$ are not equivalent

Let P be a vector space of polynomials with real coefficients. Show that the norms $ | p | _1 $ and $ | p | _2$ are not equivalent, where $|p|_1$=max$ \{|p(t)|$; $0\leq t \leq 1 \}$ and $|p|_2$ = max ...
3
votes
1answer
55 views

difference between weak* convergence and convergence

I am trying to prove the following: If $X$ is a finite-dimensional space, then for sequences $\left\{x_n\right\}\subseteq X$ and $\left\{f_n^*\right\}\subseteq X^*$, if there exists an $x\in X^*$ ...