0
votes
3answers
36 views

Let $T : \mathbb{R}^3 \to \mathbb{R}^3$ be a linear trasformation. Find $T(x)$

Let $T : \mathbb{R}^3 \to \mathbb{R}^3$ be a linear trasformation with $T \left(\begin{bmatrix} 1 \\ -2 \\ -1 \\ \end{bmatrix}\right) = \begin{bmatrix} 1 \\ -1 \\ 2 \\ ...
1
vote
1answer
18 views

Is the inverse of a symmetric positive semidefinite matrix also a symmetric positive semidefinite matrix?

If we let $$S_{++}^n(\mathbb{R})$$ denote the set of all square symmetric positive definite matrix over the real numbers, then is it true if $A\in S_{++}(\mathbb{R}) \implies A^{-1} \in ...
3
votes
1answer
46 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 ...
0
votes
0answers
17 views

any theory about determinant as a function of the elements of a matrix

Let $X=(x_{ij})_{1\le i,j\le n}$ be a n by n matrix where $n\ge3$. Consider the function $f:(R^n)^n\to R$ given by the formula $f(X):=\det(X)$. (a) Assume that $rank(X)=n-1$. Is it true that X is a ...
1
vote
2answers
32 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
26 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 ...
1
vote
1answer
33 views

To prove that these matrices are invertible

Let $A$ and $B$ be $n \times n$ matrices such that $||I - AB|| < 1$. Prove that $A$ and $B$ are invertible, and $$A^{-1} = B \sum\limits_{k=0}^{\infty} (I - AB)^k \text{ and } B^{-1} = ...
0
votes
2answers
31 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 ...
2
votes
1answer
21 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 ...
0
votes
1answer
19 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
44 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 $x\in X^*$ ...
0
votes
1answer
25 views

showing that a sequence converges in the dual space of a normed vector space

Suppose that $S=\left\{s_\alpha: \alpha \in A\right\}$ is a set of points in a normed vector space $X$ such that $\overline{span}(S)=X$. If $\left\{f_n\right\}$ is a bounded sequence in $X^*$ and ...
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 ...
3
votes
2answers
123 views

Smooth spectral decomposition of a matrix

Let $A : x \mapsto A(x)$ be a $C^\infty$ map from the half-plane $\left\{ (x_1,x_2,\cdots,x_n) \in \mathbb{R}^n,\ x_n>0\right\}$ to the space of symmetric matrices with real coefficients. Suppose ...
1
vote
2answers
37 views

Example- $l_p$ norm space

$$||x||= {[{\sum _{i=1}^\infty |x_i|^p}]}^{1/p}$$ Is a norm on $l_p$ space :- space of all sequences made of scalars from $\mathfrak C$(filed of complex numbers). To prove that above is norm on $l_p$ ...
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
1answer
32 views

Understanding a Proof for Why $\ell^2$ is Complete

Setting: Let $(x_n)$ be Cauchy in $\ell^2$ over $\mathbb{F} = \mathbb{C}$ or $\mathbb{R}$. I'm trying to show that $(x_n) \rightarrow x \in \ell^2$. That is, I'm trying to show that $\ell^2$ is ...
3
votes
1answer
94 views

How to solve this system of 3 equations with 3 variables?

I stumbled upon this system with constants $a_{i,j}>0$ that I want to solve for $x,y,z \in\mathbb{R}$: \begin{align} a_{2,1}y+a_{3,1}z=& x(y+z) \\ a_{1,2}x+a_{3,2}z=& y(x+z) \\ ...
2
votes
0answers
55 views

Exterior power of a space of maps $(\mathbb{K}^T)$

We are given a set $T \neq \emptyset, \ \ p \ge 1, \ \ p_i : T \rightarrow \mathbb{K}$ Could you help me prove that if $ \phi: (\mathbb{K}^T)^p \ni (f_1, ..., f_p) \rightarrow \rho \in ...
1
vote
2answers
38 views

Eigenvalues and eigenvectors of an operator

I have $Ku(t)=\int_0^1 G(t,s) u(s) ds, u\in L^2[0,1]$ where $$G(t,s)=\begin{cases} s(1-t)~ 0\leq s\leq t\leq 1\\ t(1-s)~ 0\leq t\leq s\leq 1\end{cases}$$if the eingenvalues of $K$ are $1/k^2\pi^2$ ...
2
votes
2answers
26 views

clarification on a question about showing that the closure of a subspace is a subspace

In a homework problem, I have been asked to prove the following "If $X$ is a normed linear space and $S$ is a linear subspace of $X$ then $\overline{S}$ is a linear subspace of $X$." ($\overline{S}$ ...
1
vote
2answers
115 views

Showing that $A=B+\alpha \cdot I$ is an invertible matrix

Let $B$ be a non-zero random $n\times n$ matrix generated using the matlab command $B=rand(n,n)$. I need to show that $A=B+\alpha \cdot I$ is an invertible matrix, where $\alpha=\|B\|_{\infty}$. I ...
3
votes
3answers
72 views

Necessity of completeness of the inner product space in Riesz representation theorem

I wanted to find a counter example to show that the completeness of the inner product space is necessary in Riesz representation theorem. Please give an example of a bounded linear functional $T$ on ...
0
votes
0answers
39 views

Evaluation map, a bounded linear functional?

Let X be the space of all polynomial functions on [0,1] over the field of complex numbers, considered as a subspace of Lebesgue space L_2. Let z be a fixed complex number. Is evaluation map at z, a ...
0
votes
1answer
28 views

Direct sum decomposition of $l_2$

Let $X=(V,E)$ be a finite graph and a linear operator $\nabla: l_2(V) \to l_2(E)$ given by the formula $(\nabla f)(x,y)=$ \begin{cases} f(x)-f(y) &d(x,y)=1\\ 0 &\text{otherwise} ...
3
votes
1answer
48 views

Does having a zero eigenvalue preclude a matrix from being indefinite?

If a $3\times3$ matrix has a positive eigenvalue, a negative eigenvalue, and a zero eigenvalue, is it then, by definition, indefinite? I think so, since the matrix has both a positive and a negative ...
4
votes
2answers
70 views

How can we construct a differential equation from a system of differential equation?

Suppose we have a linear differential equation of order $n$. All of us know how to write it down as a system of linear differential equation as $X' = A_{n \times n} X_{n \times 1}$. My question is ...
0
votes
1answer
45 views

Strictly positive solution of linear equations

Suppose $A\in\mathbb{R}^{m\times n}$, $b\in\mathbb{R}^m$, and $b\in \mathcal{R}(A)$. Show that there exists an $x$ satisfying $x \succcurlyeq 0$, $Ax = b$ if and only if there exists no ...
3
votes
0answers
51 views

Deducing that polynomials span

Let us say that we are dealing with a countable family of polynomials with real coefficients in $n$ indeterminates that commute. Are there any known/common nice systematic ways to tell if their span ...
2
votes
1answer
92 views

Linear transformation that does this

I am looking for a linear transformation that transforms $$ \sum_{i=1}^3 \frac{\partial^2}{\partial x_i^2} +\Big( b_1 (x_1-x_2)^2+ b_2 (x_2-x_3)^2+b_3(x_3-x_1)^2 \Big)$$ into something that looks ...
1
vote
1answer
41 views

Existence of a nonzero vector to form

Let $ f: \mathbb {R}^m\times \mathbb {R}^m \rightarrow \mathbb {R}^m $ an alternate form of grade two. If $ m $ is odd, prove that there exists $ v\neq 0 $ such that $ f (u, v) = 0 $, for all $ u \in ...
1
vote
0answers
42 views

How to find a hilbert basis of a given subspace considering a given inner product

Let $X$ be the space of continuous functions on $[-1;1]$ to $\mathbb{R}$ with the inner product: $$\langle f,\ g\rangle = \int_{-1}^{1} \! f(x)g(x) \, dx$$ and let $U$ be a subspace of $X$ with $U := ...
0
votes
1answer
56 views

Fredholm integral equation of first kind

I want to solve the Fredholm integral equation of first kind: $$ \int_L K(x,y)U(y)dy = f(x) $$ in these equation the function $U(y)$ is the unknown and the so-called kernel $K$ and the right hand side ...
2
votes
0answers
68 views

What would this set look like

Let $S\subseteq\mathbb{R}^{3}$ be the set of $\left(x,y,z\right)$, $x\ge y\ge z$ , which are the three eigenvalues of $diag\left(1,2,3\right)+Udiag\left(-1,-2,-4\right)U^{T}$, where $U$ is an ...
0
votes
1answer
45 views

Norm of a bounded operator

Let $\phi \in C[0,1]$ and $T_{\phi}:C[0,1] \rightarrow \mathbb{R}$ be: $T_{\phi}f = \int_0^1 f(x)\phi(x)dx$ prove that $T_{\phi}$ is a continuous, linear functional and that $||{T_{\phi}}|| = ...
1
vote
1answer
35 views

Composition of continuous linear maps is also a continuous linear map

Let $V, W , X$ be normed spaces and let $T \colon V \to W$ and $ S \colon W \to X$ be continuous linear maps. show that $ S \circ T \colon V \to X$ is a continuous linear map and that $||S \circ T || ...
1
vote
1answer
36 views

Normed vector spaces and operator norm

Let $T \colon V \to W$ be a linear map between normed vector spaces $(V, \parallel \cdot \parallel_V)$ and $(W,\parallel \cdot \parallel_W)$ as $$\parallel T \parallel :={\rm sup}\ \{ \parallel T(x) ...
0
votes
0answers
36 views

Are these linear operators continuous?

For every polynomial $p(t)= \sum_{k=0}^{n} a_k t^k$ we declare its norm by $||a_k||=\sum_{k=0}^{n}|a_k|$. Now, I am supposed to check whether these maps are continuous and in case that they are I ...
2
votes
1answer
49 views

Triangle inequality in product space of normed spaces

Let $(X,||.||_X)$ and $(Y,||.||_Y)$ be normed spaces, then $||(x,y)||:=(||x||_X^p+||y||_Y^p)^{\frac{1}{p}}$ is a norm on $X \times Y$. This is absolutely clear to me, but I have troubles to verify ...
1
vote
1answer
69 views

Showing that there is always a discontinuous functional?

I am supposed to prove the following: Let $(V,||.||)$ be an infinite-dimensional space, then there is always a discontinuous function $T:V \rightarrow \mathbb{K}$ Since continuous is equivalent to ...
0
votes
1answer
117 views

What is the second frechet derivative of a bilinear map?

Calculating the first frechet derivative of a continuous bilinear map was not that hard, since you can see what the derivative is by using linearity, but what is $ f''(x,y)((a,b),(c,d))$, where f is ...
0
votes
1answer
48 views

Does it matter in functional analysis whether we know (something about) a basis?

Let's look at a few spaces in functional analysis: $(L^p,C^n([0,1]), l^p,c,c_0,d)$ I actually only know the basis of one of these spaces. Which is the one that belongs to d, given by the unit ...
1
vote
0answers
58 views

Nilpotents of order 2 are dense in the strong operator topology

I need some help with this homework problem. (The Hilbert space in question is $\ell^2(\mathbb{N})$.) I let $T \in \mathcal{B}(\mathcal{H})$ and looked at a basic nbhd in the strong operator topology ...
2
votes
1answer
146 views

How to find all surjective functions $f:M_n(\Bbb R)\to\{0,1,2,\cdots,n\}$ satisfying $f(XY)\le\min{(f(X),f(Y))}$

Let $M_n(\Bbb R)$ be the set of all real $n\times n$ matrices. Find all surjective functions $f:M_n(\Bbb R)\to\{0,1,2,\cdots,n\}$ such that $$f(XY)\le\min{(f(X),f(Y))}$$ for all $X,Y\in M_n(\Bbb R)$. ...
1
vote
1answer
302 views

Deducing that a matrix is indefinite using only its leading principal minors

$A$ is indefinite iff $A$ fits none of the above criteria. Equivalently, $A$ has both positive and negative eigenvalues. Also equivalently, $x^TAx$ is positive for at least one $x$ and ...
0
votes
0answers
94 views

Q: Constructing Matrix for a Tight Frame

Consider a frame $\Phi = \{\varphi_1, ..., \varphi_m\}$ for $\Bbb C ^n$ consisting of $m \geq n$ vectors. Denote the frame bounds by $A \leq B$, the Bessel map and its associated $m \times n$ matrix ...
1
vote
1answer
45 views

Show why given set is not a frame

I am rather new to this material and an explanation of what is happening would be greatly appreciated. At first glance, it seems like the sum of squares is bounded at both ends but I guess I'm ...
3
votes
0answers
102 views

Integrating the exponential of a complex quadratic matrix

Problem statement I'm trying to do a discretized path integral/functional integral. The integral that I'm stuck with is of the form $$ \int_{-\infty}^{+\infty} \mathrm{d}^3\vec{x}_1\, ...
1
vote
1answer
96 views

Suggestions for comprehensive maths book library

I've problem that I'm slowly forgetting the math I've learned in early years at university (right now I'm in final year of Mgr. degree as theoretical physicist). I'd like to assemble a finite but ...
0
votes
1answer
96 views

“Openning Mapping Theorem” from $\mathbb R^n$ to $\mathbb R^m$

So need to prove if $f\colon \mathbb R^n \to \mathbb R^m$ is linear and onto, then the image of any open set is open. I figured that the first step is probably showing $f$ is continuous, and then do a ...