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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1answer
58 views

a compact operator on $l^2$ defined by an infinite matrix

Let $A$ be an infinite matrix such that $\displaystyle \sum_{i,j}|a_{i,j}|^2<\infty$. Then $A$ defined a compact operator on $l^2$.
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2answers
122 views

prove that this operator is not compact

Let $g\in C[0,1]$ be a continuous function and $g\ne 0$. Let $G:C[0,1]\to C[0,1]$ the operator defined by: $G(f)(x)=f(x)g(x)$. I proved that the operator is linear and continuous. I want to prove that ...
2
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3answers
399 views

finite dimensional range implies compact operator

Let $X,Y$ be normed spaces over $\mathbb C$. A linear map $T\colon X\to Y$ is compact if $T$ carries bounded sets into relatively compact sets (i.e sets with compact closure). Equivalently if $x_n\in ...
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0answers
83 views

Help for applying Hahn Banach theorem in this exercise .

I want to use this version of Hahn Banach: $X$ real vector space, $M$ a subspace, $p$ a sub-linear mapping and $T:M\rightarrow\mathbb{R}$ linear and $T(x)\leq p(x)$ $\forall x \in M$ Then ...
2
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0answers
289 views

Prove: Completion of a normed space is a Banach space.

Me again with another (most likely) easy problem that is too hard for me as of now. Suppose we have a normed space $(X,||\cdot||)$ and a map $i:X\rightarrow \hat X$ with $\overline{i(X)} = \hat ...
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1answer
37 views

construction of normed linear spaces from inner product spaces

How do i construct a normed linear space from a inner product space and verify that what i have suggested is true, that is, a test of verification. Moreover, does the norm have to satisfy the ...
5
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0answers
125 views

Operator norm of a convolution

Consider the operator on $L^2(\Bbb R)$, $f\rightarrow f*g$, where $g\geq 0$ is some $L^1$ function. Show the operator is a bounded linear operator with operator norm equal to $||g||_1$. Showing ...
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1answer
143 views

Let $E$ be a Banach space, prove that the sum of two closed subspaces is closed if one is finite dimensional

Let $E$ be a Banach space and let $S$ and $T$ be closed subspaces, with dim$\space T<\infty$. Prove that $S+T$ is closed. To prove that $S+T$ is closed I have to show that for any limit point $x$ ...
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0answers
47 views

proving that some transpose operator of the transpose operator is not injective.

Let $X,Y$ be normed spaces over $\mathbb k$ (where $\mathbb k$ could be $=\mathbb C$ or $\mathbb R$). Let $T\in L(X,Y) $ (a continuous and linear operator) then we define $T':Y'\to X'$ by $T'y'=y'T$. ...
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1answer
105 views

Calculate the norm of this operator

$C[0,1]=\{ f : [0,1]\to [0,1], f$ continuous$\}$ $||f||_\infty=\max_{t\in [0,1]} |f(t)|$ $T:C[0,1]\to C[0,1]$ defined by $$(Tf)(t)=\int_0^1e^{s+t}f(s)ds$$ Find $||T||$ The usual way to do this ...
2
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1answer
148 views

Functional analysis, help in Hahn-Banach theorem application

$M$ is the subspace of $L_p[a,b]$ that $\forall f\in L_p[a,b]$ $\exists g\in M$ with $f(t)\leq g(t)$ almost everywhere. $T:M\rightarrow \mathbb{R}$ $\quad$$T(f)\geq0$ whennever $f(t)\geq0$ a.e ...
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0answers
26 views

Norm of the maximum

Consider the norm $||f||= max_{x\in[a,b]} |f(x)|$ defined in the bectorial space $C[a,b]$ I have to what is the meaning (/interpretation) in $R$ of {$||f_n-f||$}$\to$ 0 Could you help me?
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1answer
38 views

Let V be a normed vector space. Show that its norm is induced by a scalar product if and only if it satisfies the parallelogramm inequality.

I managed to prove left to right, but I found it hard to get the other direction even if I have the polarization identity. I've found Fréchet - Von Neumann - Jordan theorem that proves exactly what ...
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0answers
140 views

Two norms $\|\cdot\|_a$ and $\|\cdot\|_b$ on $X$, and a function $f:X\to Y$ Fréchet differentiable with one of the norms but not with the other one?

There is a theorem that if $f : (X,\|\cdot\|_{X1}) \to (Y,\|\cdot\|_{Y1}) $ is Fréchet differentiable, then replacing the norms with some equivalent norms $\|\cdot\|_{X2}$ and $\|\cdot\|_{Y2}$ ...
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1answer
27 views

the set of points that are annhilated by a subset $N\subset X'$

Let $X$ be a normed vector space over $\mathbb k$ ($\mathbb k=\mathbb R$ or $\mathbb C$). Let's consider $X'$ the set of continuous linear functionals $f:X\to \mathbb k$ called the dual of $X$. We ...
5
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1answer
427 views

Dual to the dual norm is the original norm (?)

I have the following questions about dual norms : How do you prove that the dual of the dual norm is in fact the original norm? This is what I have so far: If I have $\|y\|_* $ as the norm dual of ...
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2answers
53 views

normed space little exercise

$(X,K)$ a normed space $E$ is a subspace of $X$. if $\exists x_0 \in X$ that $||x_0||=d(x_0,E)=1$ then show $$||e+\lambda x_0||\geq\frac{||e||}{2}\quad \forall e\in E \quad \forall\lambda\in K $$
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2answers
97 views

Norm space, linear operator exercise, help please!

$f \in L_2[a,b]$ $Uf(s):=\int_a^bk(s,t)f(t)dt$ $k(s,t):[a,b]^2\to R $ continuous. show 1) $U:L_2[a,b]\to L_2[a,b]$, in other words, $Uf(s)\in L_2[a,b] \quad \forall f$ 2) $U$ is linear and ...
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1answer
62 views

Show that the sequence converges to 0 under any norm in the space (R,‖.‖) [closed]

Show that the sequence $a_n = 1/n^2$ converges to 0 under any norm in the space $(\mathbb{R},\left\| \cdot \right\|)$.
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2answers
498 views

Show that $f(x)=||x||^p, p\ge 1$ is convex function on $\mathbb{R}^n$

Show that $f(x)=||x||^p, p\ge 1$ is convex function on $\mathbb{R}^n$. I have tried to use Holder's inequality, but I still cannot solve this problem. Could you help me with this problem? Thank you ...
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1answer
46 views

Help in proving that a vector norm satisfies an axiom.

I am trying to prove if the following is a vector norm: ||x|| = max{$|x_1 + x_2|, |x_2 + x_3|, |x_3 + x_1$|} (x is vector with 3 elements) I'm stuck proving that $||\alpha x||=|\alpha|*||x||$. I ...
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1answer
72 views

Conditional number: exercise

Let's say we've got a $202 \times 202$ matrix $A$ for which $||A||_2=100$ and $||A||_F=101$ (the Frobenius norm). How can we find the sharpest bound (lower) on the 2-norm condition number of $A$? ...
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1answer
167 views

prove that norm 1 of a function is less than or equal to (b-a) infinity norm of that function where X=C[a,b]

Prove that norm 1 of a function is less than or equal to (b-a) infinity norm of that function where X=C[a,b] is the continuous space function f:[a,b] to R. norm 1 is defined to be ntegral of the ...
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1answer
19 views

how to check wheather the given normed sequence is finite or not

Does the sequence $$ (\frac{1}{\sqrt 3}, \frac{1}{\sqrt 8},\ldots,\frac{1}{\sqrt{n^2-1}},...) $$ have finite $\ell^2$ norm? I have tried this problem using the definition of $\ell^2$ space. Is it ...
9
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1answer
95 views

Inner product space, points cannot be placed inside a ball of a given radius

I've found a very nice problem and I don't know how to go about solving it. Let $(E, || \cdot ||)$ be an inner product space, $x_1, ..., x_n \in E$. Prove that if for $i \neq j$ we have ...
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2answers
90 views

Continuous iff composition with every linear functional is continuous

Let $X$ and $Y$ be normed spaces, $T:X \rightarrow Y$ a linear operator. Show that $T$ is continuous if $y' \circ T$ is continuous for every $y' \in Y'$. My idea is the following: Suppose $T$ is not ...
2
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1answer
146 views

uniform convergence on compact subsets of the linear,continuous and uniformly bounded operators.

Let $X,Y$ be normed spaces. Let $T_j : X\to Y$ be a sequence of linear and continuous functions, such that $\lVert T_j\rVert\lt K$ $\forall j$. If $T_j$ converges pointwise to $T$, prove that $T$ is ...
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0answers
52 views

Show that $S_x$ contains an interval

$X$ is Normed vector space, and $S_x={\{x\in X\space|\space\space\space||x||=1\}}$. Prove that if exist $x,y\in X$ which are linearly independent such that $||x+y||=||x||+||y||$, hence exist $u,v\in ...
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1answer
27 views

Norms on $\mathcal{M}_{n \times n}(\mathbb{R})$

On the space of all matrices $n \times n$ with real coefficients $\mathcal{M}_{n \times n}(\mathbb{R})$ we define two norms: $||A||_1 := sup_{x\neq 0; x \in \mathbb{R}^n} \frac{||Ax||}{||x||} \\ ...
4
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2answers
165 views

Show that $F+G$ is closed when $G$ a closed subspace of normed space $E$ and $F$ a finite dimensional subspace of $E$.

Question: Let $E$ be a normed space. Let $G$ be a closed subspace of $E$ and let $F$ be a finite dimensional subspace of $E$. Show that $F+G$ is a subspace of $E$ and is closed. I'm having trouble in ...
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1answer
62 views

Series constructed from a cauchy sequence

Given a cauchy-sequence $\{x_i\}_{i\in \mathbb{N}}$ in a normed space $X.$ I need to construct a series that converges in $\mathbb{R}$ with $\{y_i\}_{i\in \mathbb{N}}$ a sequence in $X$: ...
2
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1answer
399 views

partial derivatives continuous $\implies$ differentiability in Euclidean space

I am given this theorem: If $f \in C^1(A,\mathbb R^m)$, i.e. every partial derivative of $f$ is continuous on $A$, and $A$ is open in $\mathbb R^n$, then $f$ is differentiable on $A$. Is the ...
4
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1answer
97 views

Additive norm $||a+b||=||a||+||b||$

I've read somewhere that there exist spaces where $||a+b||=||a||+||b||$ is true iff $a = \lambda b, \ \ \lambda>0$. Could you tell me what spaces have that property and what spaces don't? ...
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1answer
2k views

The definition of locally Lipschitz

I am given this definition: A function $f:A\subset\mathbb R^n\to\mathbb R^m$ is locally Lipschitz if for each $x_0\in A$, there exist constants $M>0$ and $\delta_0 >0$ such that ...
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2answers
156 views

If a sequence of summable sequences converges to a sequence, then that sequence is summable.

Let $(a_i)^n$ be a sequence of complex sequences each of which are summable (they converge). Then if they have a limit, the limit sequence $(b_i)$ is also summable. All under the sup norm for ...
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1answer
36 views

In a normed space, characterization of the sets $F$ such that $d(x,F)$ is achieved for all $x$?

Let $(E,\|.\|)$ be a real normed vector space and $d$ be the distance associated to the norm. I am wondering if there exists a characterization of the subsets $F$ of $E$ such that for all $x\in E$, ...
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1answer
398 views

Is this valid: Every Cauchy sequence in a normed space is absolutely convergent.

Proof. Let $X$ be a normed space with norm $|\cdot |$ and $(x_n)$ be Cauchy. Then for all $\epsilon \gt 0, \ \exists N : m,n \gt N \implies |x_m - x_n| \lt \epsilon$ is the standard definition of ...
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0answers
62 views

Codimension in X, given the codimension in a subspace of X

Let $X$ be a normed space with infinite dimension, let $M$ be a dense subspace of $X$ with codimension $k$ and let $N$ be a dense subspace of $M$ with codimension $l$ in $M$. The codimension of $N$ ...
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1answer
93 views

Trouble in proving that $\|x\|_p = \max|x_j|$

We define p-norm in this way: $\|x\|_p = \{\sum ^N_j_=_1|x_j|^p\}^ {1\over p}$ We know that It change to $\|x\|_p = \max|x_j| $ when $ p \to \infty $ How can I prove this ?
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1answer
462 views

Is every normed vector space a metric space?

I was trying to prove that every normed space is a metric space, and the first three proprierties came natural. However, when faced with proving the triangle inequality I had a bit of problems. I ...
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1answer
61 views

existence of a weakly cauchy sequence if the dual space is separable [closed]

Let X be a normed space such that $X^*$ is separable. Given any sequence $(x_n)\in X$ then there exist a subsequence weakly cauchy
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1answer
88 views

Isometry from Banach Space to a Normed linear space maps

Let $X$ be a Banach Space and $Y$ be a normed linear space. Show that if $T$ is an isometry then $T(X)$ is closed in $Y$. Let me have some idea to solve this. Thank you for your help.
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1answer
176 views

Proving that if $f$ is not continuous functional then $\ker f$ is dense

In the context of a first course in functional analysis I have seen the following exercise: Let $X$ be a normed space and $0\neq f$ a functional. Prove that if $f$ is not continuous then $\ker ...
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1answer
84 views

Finiteness of the dimension of a normed space and compactness

I am studying functional analysis, and in the setting of normed spaces I have seen the theorem that states that the unit ball is compact iff the space is finite dimensional. I also saw an exercise: ...
2
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1answer
118 views

Proving that $X/M$ is a Banach space when $X$ is

I am trying to do an exercise in an introduction to functional analysis course: 1) Let $X$ be a normed space and $\{x_{n}\}_{n=1}^{\infty}\subseteq X$. Prove that $X$ is a banach space iff ...
5
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2answers
106 views

Linear operators on $C^\infty[a,b]$

I do not know too much about linear operators so forgive me if this doesn't make much sense, but what would the space of linear operators on $C^\infty[a,b]$ be defined as? If we denote this space as ...
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1answer
112 views

Lipschitz condition in infinite dimensional vector spaces

If we have that $T:V \times W \rightarrow Y$ multilinear and $V,W$ are infinite-dimensional normed vector spaces.(the finite-dimensional proof is easy, since you can use compactness of the boundary ...
4
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2answers
498 views

Is the boundary of the unit sphere in every normed vector space compact?

I wanted to ask whether the boundary of the unit sphere in every normed vector space is compact? I know that this is true for simple examples, but how is it in general?
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1answer
29 views

Clarification about the space $C^0 ([-T,T],B)$

Let $B$ be a Banach space (in particular, $B$ is a function space equipped with the supremum norm). The space $C^0 ([-T,T],B)$ is the set of continuous functions on $[-T,T]$, valued in $B$. ...
0
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1answer
121 views

a normed vector space is normed closed iff it is weakly closed.

The claim is A subspace of a normed vector space is normed closed iff it is weakly closed. I can show one direction. Strong convergence implies weak convergence, so it is weakly closed. But I have ...