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 ...
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 ...
0
votes
1answer
39 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
46 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
56 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 ...
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 ...
0
votes
2answers
43 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
38 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
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 ...
2
votes
0answers
72 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
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 ...
1
vote
3answers
53 views

Find a value $c$ such that $\left\|\begin{pmatrix} x^2 - y^2\\2xy \end{pmatrix}\right\| \leq |c|\left\|\begin{pmatrix} x\\y \end{pmatrix}\right\|$

This is part of a larger problem where I am trying to find the derivative of a vector valued function. I feel like I'm missing something simple. NOTE: $c$ can be a function of $x$ and $y$.
0
votes
2answers
61 views

Eigenvalues of $d/dx$.

Consider $d/dx:C^\infty(\mathbf{R})\rightarrow C^\infty(\mathbf{R})$ (both as real vector spaces). I want to find its eigenvalues and corresponding eigenvectors. Every $\lambda\in\mathbf{R}$ is an ...
1
vote
1answer
38 views

show that for any $n \in \mathbb{Z}$ gcd($n^2 - n + 1, n +1)$ is either $1$ or $3$.

show that for any $n \in \mathbb{Z}$ gcd($n^2 - n + 1, n +1)$ is either $1$ or $3$. My Work: I considered the case where $n =-1$ , and the case $n \not= 1$. So when $n\not= -1$ we can let $n^2 - ...
0
votes
2answers
56 views

Solution in general for a seemingly simple problem

Let $\mathbb{S}$ be a closed, bounded, convex set in $\mathbb{R}^N$. Let $\mathbb{x}=[x_1,\dots,x_N]$ be any arbitrary vector in $\mathbb{S}$. Then what can we comment on the problem \begin{align} ...
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 ...
0
votes
1answer
11 views

Nonnegative linear recurrent sequence plus roots with modulus $>1$ implies goes to infinity?

Let $(u_n)_{n\in\mathbb Z}$ be a nonzero sequence of nonnegative real numbers satisfying a linear recurrence with constant coefficients : $$ u_{n+r}=\sum_{j=0}^{r-1} a_ju_{n+j} \ (n\in {\mathbb Z}) ...
0
votes
1answer
32 views

Is any norm $||\cdot||:V\rightarrow [0,\infty)$ surjective?

Let $V$ be a vector space over $\mathbb{R}$ or $\mathbb{C}$. Let $||\cdot||$ be a norm on $V$. I know that when $V$ is nonzero finite-dimensional, $||\cdot||:V\rightarrow [0,\infty)$ is surjective. ...
0
votes
1answer
26 views

Prove that the operator norm is a norm

Exercise: Prove that the operator norm of the set $S$ of all linear operators $L:R^n\to R^m$ defines a norm on $S$ Definition of norm: A positive function $\| .\|$ on a real vector space $V$ is a ...
0
votes
0answers
29 views

linear continuation

I want to ask something about the following setting Let $F$ be a normed linear space, $E\subseteq F$ a subspace and $G$ a Banach space. Let $T: E \rightarrow G$ a bounded linear map. I have to prove ...
1
vote
2answers
67 views

Homogeneity and Differentiability at $0$ implies linearity?

Suppose $f: \mathbb{R}^n \to \mathbb{R}^m$ is homogeneous and differentiable at $0$, then does it follow that $f$ is a linear transformation? I know that I need to show that for any $x,y \in ...
0
votes
1answer
55 views

How can solve this integral

How ı can solve this integral, ı thınk that ı can seperate as above ı did but I dıd not do it, thanks for helping..
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 $X$." ($\overline{S}$ ...
0
votes
1answer
59 views

Prove that $d(x,y) :\mathbb R^2\times\mathbb R^2 \to \mathbb R$ is a distance function.

$$ d(x,y) = \begin{cases} \|x-y\| & \text{if }x\text{ and }y\text{ are linearly dependent} \\ \|x\|+\|y\| &\text{otherwise} \end{cases} $$ I am stuck. Do I use the definition of a metric ...
1
vote
1answer
36 views

Are these inner product spaces?

1) Vector space of $2\times2$ real matrices and $(A,B)=\text{trace}(AB)$ 2) Vector space consisting of all polynomials of degree $2$ with $\langle p,q\rangle=p(-1)q(-1)+p(1/2)q(1/2)+p(-1)q(-1)$ How ...
0
votes
1answer
25 views

question about normed vector spaces and its subspaces

We know that every finite dimensional vector subspace of a normed space $X$ is closed in $X$. Does the result also holds for infinite dimensional subspaces of $X$ ? MY answer is not. For instance, ...
0
votes
1answer
27 views

Inequality matrix norm

Let $A$ be an $n\times n$ random matrix $A=rand(n,n)$. Let $\alpha=max_{i,j}|a_{ij}|$ (i.e, $\alpha$ is the largest entry in $A$ in absolute value).I need to show that $\ \alpha < \| A \|_{2}$. ...
1
vote
0answers
69 views

Linear transformations are Lipschitz and continuous

I'm a little confused about the proofs that linear transformations $f:\mathbb{R}^n \to \mathbb{R}^n$ are a) continuous and b) Lipschitz. I know that Lipschitz implies continuity. However, the only ...
1
vote
1answer
38 views

Orthogonal complement in pre-hilbert space

I just want to be sure that the following is correct: Let $T:H \rightarrow H$(continuous), where $H$ is a pre-Hilbert space, then we have $H=\ker(T) \oplus\ker(T)^{\perp}$, where $\ker(T)^{\perp}$ is ...
4
votes
2answers
83 views

Finding the Integer part of $\sum_{k=2} ^{9999}\frac{1}{\sqrt k}$

Question from Model Question Paper for B.Math/B.Stat: Page 28, Question 27 by Indian Statistical Institute Q. Assume the following inequalities for positive k: $$\frac{1}{2\sqrt{k+1}}< ...
0
votes
4answers
90 views

How can I show that an odd degreed polynomial with coefficients in the real space always has a root in $\mathbb{R}$? [closed]

How can I show that every odd degreed polynomial with coefficients in the real space will have a real root?
0
votes
1answer
21 views

What does it mean for $\max_{1\leq{i}\leq{n}}x_i-1<\theta<\min_{1\leq{i}\leq{n}}x_i$?

Suppose we have that $x_1,...,x_n$ are values $\in \mathbb{R}$. Here is the following problem: 1) How is it possible to have $\max_{1\leq{i}\leq{n}}x_i-1<\theta<\min_{1\leq{i}\leq{n}}x_i$? ...
1
vote
0answers
42 views

Properties of matrix functions

Can I say that a certain matrix function is absolutely continuous or monotonically increasing in $\lambda$(assuming that the matrix is a function of $\lambda$)? In other words, are these ...
0
votes
1answer
30 views

Let $f: M_{m\times n} \to M_{n\times n}$ be given by $f(A) = A^tA$. Prove that $f$ is differentiable and find a formula for $f'(A)$

Let $f: M_{m\times n} \to M_{n\times n}$ be given by $f(A) = A^tA$. Prove that $f$ is differentiable and find a formula for $f'(A)$. I am a little confused on this problem. First of all, the ...
2
votes
4answers
35 views

If $ f:\mathbb{R}^n \to \mathbb{R}^n$ is an involution, then there exists a basis $\{v_i\}$, such that $ f(v_i) = v_i $ or $ f(v_i) = -v_i$?

If $ f:\mathbb{R}^n \to \mathbb{R}^n$ is an involution, then there exists a basis $ {v_1,...,v_n } $ for $ \mathbb{R}^n$such that for each $ i $ $ f(v_i) = v_i $ or $ f(v_i) = -v_i$? This statement ...
-6
votes
2answers
118 views

sufficient and essential condition for $P(x)$ and $Q(x)$, such that $P(\sin x)= Q(\cos x)$ [closed]

What is the sufficient and essential condition for two real polynomials $P(x)$ and $Q(x)$, such that $P(\sin x)= Q(\cos x)$ for $x\in (\alpha, \beta)$, $\alpha\lt \beta$?
4
votes
2answers
93 views

Show that if a function $f : \mathbb{R}^n \to \mathbb{R}^m$ is differentiable with differentiable inverse then $m = n$

So far I have: $\boldsymbol{f^{-1}} \circ \boldsymbol{f}(\boldsymbol{a}) = \boldsymbol{a} \implies [\boldsymbol{D}(\boldsymbol{f^{-1}}(\boldsymbol{a}) \circ \boldsymbol{f}(\boldsymbol{a}))] = I_n ...
1
vote
1answer
71 views

If the vector space of all real valued continuous functions on the metric space (X,d) is finite dimensional then X is finite set

If $(X,d)$ is a metric space such that $C(X,R)$ is a finite dimensional real vector space, would any one help me to show that $X$ is finite set? $C(X,R)$ denotes the set of all real valued continuous ...
0
votes
1answer
30 views

How do I prove a bound on the sum of volumes of disjoint rectangles in an oblique rectangle?

Before I start, let's not assume anything about volume, since it is not precise unless Lebesgue measure is defined. Let $\mathscr{B}=\{\prod_{i=1}^n [a_i,b_i)\subset \mathbb{R}^n:a_i\le b_i\}$ ...
1
vote
1answer
42 views

How do i prove this? (I'm not sure what exactly the title should ne)

Let $R=\prod_{i=1}^n [0,a_i]$ and $Q=\prod_{i=1}^n [c_i,d_i]$ in $\mathbb{R}^n$ Let $T:\mathbb{R}^n \rightarrow \mathbb{R}^n$ be an orthogonal operator and $x=(c_1,...,c_n)$ Say, $Q\subset T(R)$. ...
0
votes
2answers
35 views

Determine polynomials with $n$-variables

Here is a funny problem arise from harmonic analysis: Let $E$ be a measurable subset of $\mathbb R^n$ with $m(E)>0$, where $m$ is the usual Lebesgue measure on $\mathbb R^n$. In practice, $E$ ...
0
votes
2answers
33 views

Question on inequalities. How to show if $a+b+c+d+e\le r+s+t+u+v$ and $a-r\ge0, b-s\ge0,c-t\ge0,d-u\ge0,e-v\ge0$ implies $a=r, b=s,c=t,d=u,e=v$

Question on inequalities. How to show if : $a+b+c+d+e\leq r+s+t+u+v$ and $a-r \geq 0, b-s \geq 0,c-t \geq 0,d-u \geq 0,e-v \geq 0$, then $a=r, b=s,c=t,d=u,e=v$. How do you show this is true. I can ...
0
votes
1answer
86 views

Linear Operator and isomorphism

I wanted to be sure about the following: Let's say we have vector spaces normed spaces $X$ and $Y$ and a linear operator $T:X \rightarrow Y$. My idea was to reduce the properties that I need to show ...
1
vote
0answers
65 views

Can we list all the orthonormal bases of $C[0,1]$?

Let $C[0,1]$ denote the set of all real valued continuous functions over $[0,1]$. Can we list all the orthonormal bases of $C[0,1]$? In particular my interest to know that does there exist any basis ...
4
votes
2answers
103 views

Gram-Schmidt in Hilbert space?

EDIT: After some contemplation I decided to phrase the question better to avoid trivial answers. Consider a Hilbert space with a basis $\{v_{i}\}$ where $i\in I$ an index set, which could be ...
1
vote
1answer
44 views

Why does this subspace need to be closed?

I am trying to prove the following: Let $U$ be some closed subspace and $x \in X$ but not in $U$, then there is a functional such that $x'|_U=0$ and $x'(x) \neq 0$. My idea was the following: Let ...