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.

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28 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 ...
1
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0answers
120 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{...
1
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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 {...
0
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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 $\|\cdot \|$?
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1answer
83 views

What is the relation between the matrix of a bounded linear operator and that of its adjoint?

Let $X$ and $Y$ be finite-dimensional normed spaces, both real or both complex, and let $\dim X = n$ and $\dim Y = m$. Let $E \colon= ( e_1, \ldots, e_n )$ be an ordered basis for $X$, and let $F \...
3
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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 ...
0
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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 ...
1
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1answer
44 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
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4answers
79 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{...
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0answers
298 views

Is it possible to characterize completeness of a normed vector space by convergence of Neumann series?

If $X$ is a normed vector space and if for each bounded operator $T \in B(X)$ with $\| T\| < 1$, the operator ${\rm id} - T$ is boundedly invertible, does it follow that $X$ is complete? Context: ...
2
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1answer
41 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
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1answer
231 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{...
12
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1answer
208 views

In a real normed linear space if $||x||=||y||$ implies $\lim_{n \to \infty} ||x+ny||-||nx+y||=0$ , then the norm comes from an inner-product space?

$(V,\|\cdot|)$ be a real normed linear space such that $\|x\|=\|y\|$ implies $\lim\limits_{n\to\infty} \|x+ny\|-\|nx+y\|=0$, then is it true that the norm comes from an inner-product space ?
1
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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
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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 $T\in B(X,Y)$
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0answers
157 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
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0answers
35 views

Continuous function with support continuously embedded [duplicate]

Can someone give me a solution for this? We define $$C^1_c(\mathbb R_+) := \{f \in C^1(\mathbb R_+): \overline{\{x \in (\mathbb R_+) : f(x) \neq 0\}} \text{ compact in } (\mathbb R_+,|\cdot|)\}$$ ...
0
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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: ...
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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,...
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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 ...
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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
3
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2answers
166 views

Why is the image of a compact operator separable?

Let $A$ and $B$ be normed vector spaces and let $S\in \mathscr{K}(A,B)$ be a compact operator. Question: How does it follow that the image of $S$ is separable? Thanks for the help.
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1answer
63 views

Application of Hahn Banach Separation theorem

I am solving an exercise (not Homework).. Let $E_1$ and $E_2$ be non empty disjoint convex subsets of $X$, with $E_1$ compact and $E_2$ closed in $X$. Then there are $f\in X'$ and $t_1,t_2$ in $\...
0
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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 ...
3
votes
2answers
128 views

Prob. 6, Sec. 4.3 in Kreyszig's Functional Analysis Book: What are all possible extensions?

Here's Theorem 4.3-2 in Introductory Functional Analysis With Applications by Erwine Kreyszig: Let $X$ be a normed space, let $Z$ be a subspace of $X$, and let $f$ be a bounded linear functional ...
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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
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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'...
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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_\...
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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 ...
2
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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\...
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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)...
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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
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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 \...
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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 $\theta,\phi \in \mathbb{R}$ $f(x,y) \leq ||x||^2 + ||y||^2 - 2Re (\langle x,y \rangle )$ ...
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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 ...
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0answers
16 views

Norm vs A-norm in non-Archimedean Functional Analysis

Let $K =(K,| \cdot |)$ be a non-Archimedean valued field. Let $E$ be a $K$-vector space. A norm on $E$ is a map $||\cdot||:E\to[0,\infty)$ such that: $||x||=0$ if and only if $x=0$, $||\lambda x||=...
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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 ...
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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
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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
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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 $1$ ...
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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 ...
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,...
5
votes
1answer
830 views

Equality in Young's inequality for convolution

I am trying to understand how sharp Young's inequality for convolution is. The inequality says $||f \ast g||_r \leq ||f||_p ||g||_q$ where as $1/p+1/q = 1+1/r$. Actually, there are a couple of papers ...
1
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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\...
1
vote
2answers
555 views

continuity in the strong topology implies continuity in the weak one

I have to prove that if $T:(E,\|\cdot\|_E)\rightarrow (F,\|\cdot\|_F)$ is a continuous and linear operator, and $x_h\rightharpoonup x$ in $E$, than $Tx_h\rightharpoonup Tx$ in $F$. So we know that $T$ ...
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
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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 ...