A vector space $E$, generally over the field $\mathbb R$ or $\mathbb C$ with a map $\lVert \cdot\rVert\colon E\to \mathbb R_+$ satisfying some conditions.

learn more… | top users | synonyms

0
votes
1answer
19 views

element-wise order implies norm order?

Let $v_1, v_2 \in \mathbb R^n$. If $0\le v_1 \le v_2$ element-wise, is it true that $\|v_1\| \le \|v_2\|$ for any norm $\|\cdot \|$?
1
vote
0answers
99 views
+50

Linear transformation $T$ such that for every extension $\overline{T}$, $\|\overline{T}\|>\|T\|$.

Let $E$ and $F$ be normed spaces such that $\dim F < \infty$, $G$ a subspace of $E$ and $T:G\rightarrow F$ a continuous linear map. I know that there exists a continuous linear extension $\overline{...
0
votes
1answer
27 views

distance between two eigen vectors corresponding to two different matrices in a normed space

Let $A$ and $B$ are two $n\times n$ matrices. Let 1) $Ax = \lambda x$ and 2) $By=\mu y$ for $x,y$ in a normed space. $\lambda, \mu$ are scalar. Also, for $x,y$ are unique eigen vectors (upto a ...
0
votes
1answer
35 views

Variant of Holder's inequality: $\|x\|_p \le n^{\frac1p- \frac1r} \|x\|_r$

So far I believed that only the reverse Holder inequality holds for $0<p<r<1,$ but then a student pointed out to me that $$\|x\|_p \le n^{\frac{1}{p}- \frac{1}{r}} \|x\|_r.$$ A few numerical ...
3
votes
1answer
77 views

Tricky norm-inequality $\|x\|_p \le n^{\frac{1}{p}- \frac{1}{r}} \|x\|_r.$ for $p \in (0,1)$

For $r>p \ge 1$ one can show that in $\mathbb{C}^n$ we have $$\|x\|_p \le n^{\frac{1}{p}- \frac{1}{r}} \|x\|_r.$$ My question is now: Does this also hold for $1 \ge r>p>0$? Obviously we ...
1
vote
1answer
43 views

Prob. 2, Sec. 2.8 in Kreyszig's functional analysis text: What is the norm of these bounded linear functionals on $C[a,b]$?

Let $C[a,b]$ denote the normed space of all the continuous (real or complex-valued) functions defined (and continuous!) on the closed interval $[a,b]$ on the real line, where $a, b \in \mathbb{R}$ and ...
1
vote
4answers
78 views

prove triangular inequality for $ d(x,y)= \frac{||x-y||}{1+||x-y||}$ [duplicate]

prove triangular inequality for $$ d(x,y)= \frac{||x-y||}{1+||x-y||}$$ that is $d(x,y) \leq d(x,z)+d(z,y)$ ofcourse ||.|| is a norm and has properties of norms this usually works $$ \begin{...
2
votes
1answer
39 views

Norm of convolution

Let $f,g: \mathbb{R}^n \to R$. Let $|| \cdot ||_T$ be a translation invariant norm on functions on $\mathbb{R}^n$. How can I prove that $||f*g||_T \leq ||f||_1 ||g||_T$ (where $f*g$ means convolution ...
2
votes
1answer
39 views

A linear map $S:Y^*\to X^*$ is weak$^*$ continuous if and only if $S=T^*$ for som $T\in B(X,Y)$

As the title says, the question is how to prove $S:Y^*\to X^*$ is weak$^*$ continuous if and only if $S=T^*$ for som $T\in B(X,Y)$
0
votes
0answers
35 views

Continuous function with support continuously embedded [duplicate]

Can someone give me a solution for this? We define $$C^1_c(\mathbb R_+) := \{f \in C^1(\mathbb R_+): \overline{\{x \in (\mathbb R_+) : f(x) \neq 0\}} \text{ compact in } (\mathbb R_+,|\cdot|)\}$$ ...
1
vote
1answer
55 views

Show that function $\mathcal F$ is norm preserving

Fix $N \in \Bbb N$. The function $\mathcal F:(\Bbb C ^N , || \cdot || _2 )\to(\Bbb C ^N , || \cdot || _2 )$ is defined as follows: $$ (\mathcal F (x))_k := \frac 1 {\sqrt N} \sum^N_{j=1} x_j \mathrm {...
1
vote
1answer
74 views

Show map is norm-preserving and determine Eigenvalues [duplicate]

Can someone of you give me a solution for this? Let $N\in \mathbb N$. a) We define the map $\mathfrak F:(\mathbb C^N, ||\cdot||_2)\to(\mathbb C^N, ||\cdot||_2)$ by $$(\mathfrak F(x))_k := \frac{...
2
votes
1answer
230 views

Show norm preserving property and determine Eigenvalues

Can someone of you give me a solution for this? Let $N\in \mathbb N$. a) We define the map $\mathfrak F:(\mathbb C^N, ||\cdot||_2)\to(\mathbb C^N, ||\cdot||_2)$ by $$(\mathfrak F(x))_k := \frac{...
0
votes
0answers
14 views

Is $\mu A+\lambda B$ closed? [duplicate]

Let $E$ be a real normed space, $\mu>0, \lambda >0$ and $A,B \subset E$ closed and convex sets such that $0 \in A$ and $0\in B$. Is $\mu A+\lambda B$ closed? This is what I did so far: ...
0
votes
1answer
65 views

$T:\mathbb R^n \to \mathbb R^n $ be an isometry , is $T$ surjective?

Let $T:\mathbb R^n \to \mathbb R^n $ be an isometry and $T(0)=0$ , then $T$ is linear and $T(B[0,1])\subseteq B[0,1]$ so $T:B[0,1]\to B[0,1]$ is an isometry and since $B[0,1]$ is compact so $T|_{B[0,...
1
vote
1answer
43 views

$B$ be an uncountable dimensional real Banach space and $T:B \to B$ be a surjective isometry such that $T(0)=0$ , then is $T$ linear?

Let $B$ be a real Banach space and $T:B \to B$ be a surjective isometry such that $T(0)=0$ , then is $T$ linear ? I know that it is true if $B$ is a Hilbert space ( we only need an inner product ...
0
votes
1answer
36 views

In $(C[0,1],||.||_{\infty})$ , is the set $\{p(x)\in C[0,1]$, $p(x)$ is a polynomial $:\int_0^1 p(x)dx=1\}$ totally bounded?

Consider the metric space $(C[0,1],||.||_{\infty})$ , in this space , is the set $\{p(x)\in C[0,1]$, $p(x)$ is a polynomial $:\int_0^1 p(x)dx=1\}$ totally bounded ? Please help , Thanks in advance
0
votes
1answer
33 views

Let $\lVert\cdot\rVert_1,\lVert\cdot\rVert_2$ be norms on vector space $X$. Prove that they generate the same topology iff they are equivalent. [duplicate]

Note that by "generate the same topology" we mean that any set that is open with respect to $\lVert\cdot\rVert_1$ is also open with respect to $\lVert\cdot\rVert_2$ and vice versa. By "equivalent" we ...
1
vote
0answers
89 views

Continuous embeddings

Given the following exercise: We define $$C^1_c(\mathbb R_+) := \{f \in C^1(\mathbb R_+): \overline{\{x \in (\mathbb R_+) : f(x) \neq 0\}} \text{ compact in } (\mathbb R_+,|\cdot|)\}$$ (the ...
0
votes
1answer
25 views

Prove or disprove: $\lVert T\rVert=\sup_{\lVert x\rVert=1}|\langle Tx,x\rangle|$, where $H$ is a Hilbert space and $T$ is bdd linear operator.

Edit: To clarify, note that $T:H\to H$. This is a problem on an old preliminary exam in Analysis that I'm working through to prep for my own prelim. My initial thought was to disprove it, but I can'...
0
votes
1answer
22 views

Bounded linear extension of an operator whose co-domain is incomplete.

Let $X$ be a normed space and $Y$ be a Banach space. The bounded linear extension theorem states that a bounded linear operator $T : M \to Y$, where $M \subseteq X$ is dense, can be extended to a ...
-8
votes
0answers
156 views

Show that map is norm preserving and determine Eigenvalues [duplicate]

Can someone of you give me a solution for this? Let $N\in \mathbb N$. a) We define the map $\mathfrak F:(\mathbb C^N, ||\cdot||_2)\to(\mathbb C^N, ||\cdot||_2)$ by $$(\mathfrak F(x))_k := \frac{...
0
votes
2answers
36 views

Is $f_n=x^n$ weakly convergent in $(\mathscr C[0,1],\lVert\cdot\rVert_\infty)$?

This is part of an old preliminary exam in Analysis I am working through. For earlier parts of the problem I have already shown that $f_n$ does not converge in $(\mathscr C[0,1],\lVert\cdot\rVert_\...
2
votes
0answers
31 views

Completeness of 'Hardy Space' $H^2(D)$

Define Hardy Space $H^2(D)$ as a space of holomorphic functions $f$ on unit open disc $D=\{z\in\mathbb{C}:|z|<1\}$ endowed with the norm $$ ||f||^2=\sup_{0<r<1} \int_0^{2\pi} |f(re^{i\...
1
vote
0answers
24 views

Prob. 4, Sec. 4.3 in Kreyszig's functional fnalysis text: Application of the Hahn Banach Theorem

Let $X$ be a real or complex vector space, and let $p \colon X \to \mathbb{R}$ be a real-valued function satisfying $$p(x+y) \leq p(x) + p(y) \ \mbox{ for all } \ x, y \in X$$ and $$p(\alpha x) = \...
0
votes
1answer
60 views

Trouble finding a function satisfying an integral equation

I'm stuck at the last step of this exercise: b) Use the Banach fixed point theorem to show that there is a unique function $f \in C[0,1]$ for which the equation $$f(t) + \int_0^1e^{\tau+t-3}f(\tau)...
0
votes
3answers
90 views

Isometric isomorphism between $R^2$ and $R$

Can someone help me solving the following problems? $(\mathbb R^2,d_2)$ and $(\mathbb R, d_1)$, $d_2, d_1$ being the respective euclidean norms, are not isometric isomorphic, i.e. there is no ...
1
vote
1answer
22 views

If $\| \varphi\| = 1$, then $\varphi (B_E) = (-1,1).$

Let $E$ be a real normed space and $\varphi \in E'$, $\|\varphi\| = 1$. Is it true that $\varphi (B_E) = (-1,1)$ ? Clearly, if $x\in B_E$, we have $\varphi(x) \in (-1,1)$ since $|\varphi(x)| \leq \...
-1
votes
1answer
29 views

Which of following inequality holds?

$f(x,y) = _{\theta,\phi}^{{sup}}$ $ ||e^{i\theta}x - e^{i\phi}y||_{2}^2$ ,where $x,y \in \mathbb{C^2}$ and $\theta,\phi \in \mathbb{R}$ $f(x,y) \leq ||x||^2 + ||y||^2 - 2Re (\langle x,y \rangle )$ ...
0
votes
1answer
42 views

Suppose $f_n\to f$ in $L^1([0,1],\lambda)$. Prove or disprove: $\exists \{f_{n_j}\}$ such that $f_{n_j}(x)\to f(x)$ for almost every $x\in[0,1]$. [duplicate]

This is part of an old preliminary exam in Analysis I am reviewing to prepare for my own prelim. $\lambda$ is the Lebesgue measure. $f_n\to f$ with respect to the $L^1-$norm. I know that there exists ...
1
vote
0answers
21 views

Which of the inequality holds?

$f(x,y) = _{\theta,\phi}^{{sup}}$ $ ||e^{i\theta}x - e^{i\phi} y||_{2}$ ,where $x,y \in \mathbb{C^2}$ and $\theta,\phi \in \mathbb{R}$ Which of the following holds ? $f(x,y) \leq ||x||^2 + ||y||^2 ...
1
vote
1answer
19 views

Every Cauchy sequence in $\{f\in (C([0,1]),\|\cdot\|_1)\,|\,\exists a,b\in\mathbb R:f(x)=ax+b\}$ converges

I have trouble proving that, using the norm $\|f\|=\int_0^1|f(x)|\mathrm dx$, for a Cauchy sequence of functions $f_n(x)=a_nx+b_n$, the sequences $(a_n)_n$ and $(b_n)_n$ also have to be Cauchy ...
0
votes
0answers
51 views

Problem regarding isometric isomorphisms [duplicate]

I need help regarding the following two exercises: a) Show that $(\mathbb R^2, d_2)$ and $(\mathbb R,d_1)$ $d_2,d_1$ being the respective euclidean metrics, are not isometric isomorphic, i.e. ...
2
votes
1answer
28 views

Non-convergent Cauchy sequence in $\ell^1$ with respect to the $\ell^2$ norm

Let $X = \ell^1$, the set of absolutely convergent real valued sequences and let $d_2(\mathbf{x},\mathbf{y}) = \left(\sum_{k=1}^\infty |x_k - y_k|^2\right)^{1/2}.$ This is the $2$-norm on the $1$ ...
0
votes
1answer
30 views

Why do we study specifically 'normed' vector spaces?

When we study vector spaces, it is useful to define a norm on it for countless reasons. I was thinking about this recently and realised Don't all vector spaces have norms on them? If they all ...
0
votes
1answer
22 views

Let $X$ be a finite dimensional normed space. Does the algebraic dual space $X^*$ and the dual space $X'$ coincide?

I am currently studying for my Functional Analysis test and then started thinking about the following and figured it is true (if it is not true, please do tell me - I am just thinking about this to ...
2
votes
1answer
54 views

Let $X=\left\{ f:[0,1]\to\mathbb R \;\big|\; \lVert f\rVert_X<\infty,\ f(0)=0\right\}$. Show $(X,\lVert\cdot\rVert_X)$ is complete.

The following is a problem on an old Analysis preliminary exam at my institution; I'm prepping for the prelim. The problem is: Let $$X=\left\{ f:[0,1]\to\mathbb R \;\big|\; \lVert f\rVert_X<\infty,...
-1
votes
1answer
67 views

Show that the following is a bounded linear operator on $L^2(R_+)$ Calculate the adjoint operator.

The following is a question I have been working on for some time with help from my teacher. Unfortunately we have a solution but are not 100% confident with it. Some guidance on if part(s) of our ...
1
vote
0answers
49 views

Showing $\|x+y\|^2+\|x-y\|^2=2\|x\|^2+2\|y\|^2 \Rightarrow \| \cdot \| $ is induced by scalar product

I need to show the above $\forall x,y,v \in V$ , a normed vector space on $\Bbb R$. A hint was given that i should first show that $$s:V \times V \to \Bbb R ; \: \:\: s(u,v):=\frac1 4 (\|u+v\|^2-\|u-v\...
2
votes
1answer
28 views

If $A$ is bounded and dissipative, is $\lambda \mathbb 1-A$ invertible for $\lambda>0$?

Let $X$ be a Banach space and $j$ a map (not necessarily linear or anti-linear!) from $X$ to $X'$ so that $$j_x(x)=\|x\|^2 \qquad \forall x \in X \text{ and }\qquad \|j_x\|_{X'}=\|x\|$$ We say that ...
1
vote
0answers
20 views

Entrywise expression for L2 matrix norm

The matrix norm induced by the $\ell^2$ norm is known to be equal to the maximum singular value of the matrix. The matrix norms induced by the $\ell^1$ and $\ell^\infty$ norms admit simple ...
2
votes
1answer
25 views

The quotient norm on $\ell^{\infty}(\mathbb{N}) / c_0 (\mathbb{N})$ is given by $\limsup_{n \in \mathbb{N}} |x_n|$.

I try to show that the norm on the quotient space $\ell^{\infty}(\mathbb{N}) / c_0 (\mathbb{N})$ is given by $\limsup_{n \in \mathbb{N}} |x_n|$, where $x = (x_n)_{n \in \mathbb{N}} \in \ell^{\infty} (\...
1
vote
0answers
27 views

Together with the algebra of cardinal numbers, is there analysis of cardinal numbers? [closed]

Let $C$ be the collection of all cardinal numbers. Is there any norm, inner-product, metric (other than discrete metric), topology(other than discrete, co-finite topology) on $C$, which is very useful?...
3
votes
0answers
36 views

Closed map $T:X \to Y$ has closed graph?

Let $T:X\to Y$ be a linear operator between two normed vector spaces. My question is: If $T$ is a closed map (sends closed sets to closed), then is the graph of $T$ a closed set of $X \times Y$? ...
0
votes
1answer
37 views

Equivalence of statements about a linear map

I need someone to help me solve the following exercise: Let $(X, ||\cdot||_X)$ and $(Y, ||\cdot||_Y)$ be normed vector spaces over a common field $\mathbb K$ $(\mathbb R$ or $\mathbb C)$. For a ...
0
votes
1answer
41 views

Exercise 1.65 of Megginson's “An Introduction to Banach Space Theory”.

Unfortunately I do not succeed in completing the following exercise: Let $X$ be a Banach space and let $T : X \to \ell^{1} (\mathbb{N})$ be a linear operator. For each $n \in \mathbb{N}$, let $(Tx)...
2
votes
1answer
26 views

Weak/Weak* topologies compared to topologies generated by semi-norms from dense subset

The weak topology on normed linear space $X$ can be defined as being induced by semi-norms $\|\cdot\|_{x'}$, $x'\in X'$ with $\|x\|_{x'}=|x'(x)|$. Similarly the weak* topology is induced by $\|\cdot\|...
0
votes
0answers
34 views

The precise definition of Cartesian coordinate and Euclidean space?

I'd searched them for a while, but still have not found a clear and unity definition on it. The problem really confused me. What is the precise definition of Cartesian coordinate and Euclidean space? ...
1
vote
0answers
17 views

Conditions for non-normability of (nontrivial proper and nondense) subspace of non-normable space.

Lets $X$ be a non-normable topological vector space and let $Y\subset X$ be a proper subspace. Clearly if $Y$ is dense in $X$ then $Y$ must be non-normable too. Can we have that conclusion with weaker ...
-1
votes
1answer
30 views

Proof/disprove contunuity of a map [duplicate]

I need help with proving / disproving something: Look at the map $$\Phi: (C([0,1], \mathbb R), ||\cdot||_{\infty}) \to (\mathbb R, |\cdot|); \,\,\,\,\,\,\Phi(u) := \int_0^1 u²(t) dt$$ a) ...