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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3answers
51 views

Do we need to have a subsequence such that $\lim_{k\to\infty}\left\|x_{n_k}\right\|=\liminf_{n\to\infty}\left\|x_n\right\|$?

Let $(X,\left\|\;\cdot\;\right\|)$ be a normed space and $(x_n)_{n\in\mathbb{N}}\subseteq X$. Can we prove that there is a subsequence ...
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0answers
37 views

How is $\|\cdot\|_1$ defined on a finite-dimensional real vector space?

Let $V$ a normed space over $\Bbb{R}$, and let $S$ be a finite dimensional subspace. I'm trying to show that $S$ is complete, I've already seen this question has ben made, but I have a precise doubt. ...
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3answers
216 views

Is a Banach space also a metric space?

Since a Banach space is a complete normed vector space and a norm always induces a metric, a Banach space must be a metric space, right? If so, why is a Banach space defined as a complete normed ...
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2answers
168 views

Is the unit sphere in an infinite dimensional Hilbert space closed?

Is a unit sphere in an infinite dimensional hilbert space closed. By the triangle inequality it is clear that the all the limit points of the sphere are inside the closed unit ball. But I cannot ...
6
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1answer
82 views

Is there a nowhere differentiable norm on $\Bbb R^n$?

Is there a nowhere differentiable norm on $\Bbb R^n$? The non differentiable norms that I know (e.g. the one norm $\|\cdot\|_1$ and the infinity norm $\|\cdot\|_{\infty}$) are non differentiable ...
9
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2answers
125 views

Isometry group of a norm is always contained in some Isometry group of an inner product?

$\newcommand{\<}{\langle} \newcommand{\>}{\rangle} $Let $||\cdot||$ be a norm on a finite dimensional real vector space $V$. Does there always exist some inner product $\<,\>$ on $V$ such ...
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1answer
31 views

Book reference for introduction to Normed Spaces

I wanted a book reference for the study of normed spaces and linear operators. Im still not into functional analysis, but I wanted a reference of an introductory book as to start reading. What would ...
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2answers
124 views

Triangle Inequality for norm integral $\|f\|_1=(\int_a^b [|f|^2+|f'|^2]\mathsf dx)^{1/2}$.

Define $C_1^1[a,b]$ to be the space of continuously differentiable functions on $[a,b]$, with norm $$\|f\|_1=\left(\int_a^b[|f|^2+|f'|^2]\mathsf dx\right)^{1/2}.$$ Show that this is a proper ...
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2answers
62 views

Is $\ell_{\infty}$ a smooth normed space?

A normed linear space $X$ is said to be smooth if for each $x \in X$ there exists a unique functional $x^* \in X^*$ with $\|x^*\|=1$ such that $x^*(x)=\|x\|$. I know that $L_1[0,1]$ is not smooth. ...
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1answer
31 views

Compute the limit of this expression of norms:

Compute the limit, as n goes to infinity, of the quotient: $$\frac{||A^{n+2}(x)||}{||A^n(x)||} $$, given the matrix $$ \begin{bmatrix} 0 & 3 \\ -2 & 5 \\ ...
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0answers
39 views

Calculating Norms for the Transpose

How does one show that the transpose mapping $T: \mathcal{M}_{n} \to \mathcal{M}_{n}$, given by $T(a)=a^{t}$, has norm $||T||=1$ but completely bounded norm $||T||_{CB}=n$? Notation. ...
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1answer
24 views

Find two functions in $L_p(\Bbb R)$, whose product $f\cdot g$ does not belong to $L_p(\Bbb R)$.

How can I find two functions in $L_p(\Bbb R)$, with their product $f\cdot g$ not belonging to $L_p(\Bbb R)$?
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1answer
38 views

Differentiability in normed spaces

I really need a help with the following exercise: Suppose $\mathbb{E}$ and $\mathbb{F}$ are normed spaces, $A \subseteq \mathbb{E}$ is an open set, $f: A \to \mathbb{F}$ is differentiable on $A$, and ...
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1answer
50 views

Taking limits of norms of a matrix raised to the nth power:

Given a matrix $$ A = \begin{bmatrix} 0 & 3 \\ -2 & 5 \\ \end{bmatrix} $$ and a vector $x = \begin{bmatrix}1&0\end{bmatrix}$, compute ...
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1answer
44 views

Prob. 11, Sec. 2.10 in Erwin Kreyszig's book

If $X$ is a normed space and dim $x=\infty$, show that the dual space $X'$ (set of all bounded linear functionals on $X$) is not identical with algebraic dual space $X^*$ (set of all linear ...
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0answers
56 views

Prob. 10, Sec. 2.10 in Erwin Kreyszig's book

Here is my question: Let $X$ and $Y\neq\{0\}$ be normed spaces, Where dim $X = \infty$. Show that there is at least one unbounded linear operator $T:X \mapsto Y$. (use a Hamel basis). This ...
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4answers
60 views

does $\|A\| > \|B\|$, $A$ and $B$ matrices, imply that $\|Ax\| >\|Bx\|$ for all $x$ in some vector space?

I'm wondering if I can prove the first inequality in a question that I am working on, does that make the second inequality automatic? Thanks,
2
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1answer
59 views

Convergence of sequence of function in norm.

Let $1\leq p<\infty$. Suppose that $\{f_k\}$ is a sequence in $L^p(X,\mathcal{M},\mu)$ such that the limit $f(x)=\lim_{k \to \infty}f_k(x)$ exists for $\mu$-a.e. $x\in X$. Asumme that ...
1
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2answers
137 views

Prob. 6, Sec. 2.10 in Erwin Kreyszig's functional analysis.

Definition (Dual space $X'$). Let $X$ be a normed space. Then the set of all bounded linear functionals on $X$ constitutes a normed space with norm defined by $$ \left \| f \right \|= ...
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1answer
52 views

Prob. 10, Sec. 4.2 in Kreyszig's functional analysis book: There is a linear functional for every sublinear functional …

If $p$ is a sublinear functional on a real vector space $X$, then there exists a linear functional $\tilde{f}$ on $X$ such that $-p(-x) \leq \tilde{f}(x) \leq p(x)$ for all $x \in X$. How to prove ...
1
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1answer
86 views

Prob. 8, Sec. 4.2 in Kreyszig's functional analysis book: Nonnegativity of a subadditive functional outside a sphere implies nonnegativity

If a subadditive functional $p$ defined on a normed space $X$ is non-negative outside a sphere $\{ \ x \in X \ \colon \ \Vert x \Vert = r \ \}$, then how to show that $p$ is non-negative for all $x ...
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1answer
12 views

Is $\inf_{y\in Y} \|\alpha x-y\|=\inf_{y' \in Y} \|\alpha x-\alpha y'\|$

In my functional analysis homework problem I was trying to a extend a linear functional from a proper sub-space $Y$ to the whole space $X \supset Y$using the Hahn Banach theorem. In order to do that I ...
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2answers
51 views

Prob. 6, Sec. 4.2 in Kreyszig's functional analysis book: Continuity of a subadditive functional at zero implies continuity

Let $X$ be a real normed space, and let $p \colon X \to \mathbb{R}$ be a functional such that $$ p(x+y) \leq p(x) + p(y) \ \mbox{ for all } \ x, y \in X$$ and such that $$p(\theta) = 0, \ \mbox{ ...
0
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1answer
24 views

Differential of a function between two normed spaces

I have a question about the differential of a function between two normed spaces. It is a simple question about the definition. In my textbook from my university, the definition is as follows: Let ...
1
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1answer
33 views

An equation involving inner products being independent of the inner product space

Let $(X,\langle\cdot\,,\cdot\rangle_X)$ and $(Y,\langle\cdot\,,\cdot\rangle_Y)$ be nonzero inner product spaces over $\mathbb{C}$. I wish to know if the following statement is true. ...
2
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2answers
250 views

Riesz's Lemma for finite-dimensional spaces [closed]

Someone could give me an idea of ​​what happens in Riesz's lemma if the dimension of space is finite?
1
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2answers
57 views

Cauchy sequence which does not converge example.

Consider the normed space $(X, \Vert \cdot\Vert) $ where $$ X=\{ (a_n)_n \quad|\quad (a_n)_n \text{ real sequence with } \lim_{n\to \infty}a_n=0 \} $$ and $$\Vert (a_n)_n\Vert:= \sum_{n\geq ...
2
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1answer
118 views

Prob. 14, Sec. 2.10 in Erwin Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS: application to a system of equations?

Let $M$ be a non-empty subset of a normed space $X$, and let $M^a$ denote the subspace of the dual space $X'$ that consists of all those bounded linear functionals that vanish at each point of set ...
1
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2answers
117 views

Prob. 8, Sec. 2.10 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS: The dual space of $c_0$ is $\ell^1$?

Let $c_0$ be the subspace of $\ell^\infty$ consisting of all sequences of (real or complex ) numbers converging to $0$. How to prove that the dual space of $c_0$ is (isomorphic to) $\ell^1$? My ...
2
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0answers
30 views

Connected set in normed space

I have this exercise: "let $E$ be a normed space and $X\subset E$ $$X~\text{connected}~\Longleftrightarrow \forall A\subset X,~\text{such that} A\neq\emptyset, A\neq X~\text{we have}~ Fr(A)\neq ...
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0answers
36 views

Theorem 3.6-2 in Erwine Kreyszig's “Introductory Functional Analysis with Applications:” Does the converse hold if the space is not complete? [duplicate]

First, a definition: Let $X$ be a normed space. A subset $M (\neq \emptyset) \subset X$ is said to be total in $X$ if the span of $M$ is dense in $X$. Now theorem 3.6-2 in Kreyszig states the ...
1
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1answer
40 views

Can't work out if this proof is sound or not. Any ideas?

Let $V$ be a normed space over some field $\mathbb K$. I proved that $$ \overline{B_r(a)} = \{v \in V \mid \|v-a\| \le r \}$$ $\subseteq $ was easy but for the $\supseteq$ direction I am really not ...
0
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0answers
36 views

Proof of dual norm relation: $\frac{1}{q} + \frac{1}{r} = 1$

Recall: $\|\|$ is a norm in $R^n$, and its dual norm is defined as $\|z\|^*=\text{sup}_{\|x\|\leq1}z^Tx$. If $q$-norm and $r$-norm are dual norm, then we have the following relation: ...
0
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0answers
38 views

Mean Value Inequalities for vector-valued functions

Let $X$ and $Y$ be Banach spaces, and let $U\subset X$ be open. If $f\colon U\to X$ is continuously differentiable and $x,v\in X$ are such that the line segment $\ell=\{x+tv\mid t\in[0,1]\}$ lies ...
2
votes
2answers
44 views

Determine interior and boundary of a set

Let $(X,||\cdot||)$ be a normed vector space, where $$X = \big\{ (a_n)_{n \geq 1} ~~|~~ (a_n)_{n \geq 1} \text{ is a bounded real sequence }\big\}$$ and $$\|(a_n)_n\| = \sup_{n \in ...
0
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1answer
66 views

The set of infinite sequences with finitely many nonzero values is dense.

Could I get a proof to this lemma or a reference if a proof is too time consuming?
0
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0answers
36 views

What is the norm for the product of normed spaces?

Suppose $(\Omega, \Sigma ,\lambda)$ is a probability space and $X_i=L^2(\Omega,\Sigma,\lambda ,[0,1])$ with norm $||.||_{L^2}$ for all $i\in I$, $I$ is finite. Is there any natural norm for the ...
1
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2answers
50 views

“isomorphic” normed spaces and reflexivity

Let X, Y be normed spaces and suppose that there exists an bijective isometry between them. And if X is reflexive, then it is intuitively clear that Y is reflexive also. But, when I tried to prove ...
4
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1answer
56 views

Unit ball separable $\Longrightarrow$ Space separable

Given a normed space $X$ and assume it is also a locally convex space in some other topology (e.g. weak or weak* if it's a dual). Assume that the unit ball $B_X$ is separable in this topology. Is it ...
0
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1answer
30 views

Two-dimensional subespace suplementary of another one

Let $E$ be a real normed space. All subspaces $S$ of codimension $1$ (hyperplane) in $E$ is either, closed or dense. What do say about a similar property when $S$ is of codimension $2$?
2
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1answer
94 views

Normed space where unit ball's weak and norm topology coincide?

Let $X$ be a normed space. Are there any infinite-dimensional examples such that the $(B_X, w\restriction_B) = (B_X, \|\cdot \|\restriction_B)$. In finite dimensions this obviously holds always true. ...
11
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2answers
445 views

Can the sum of two measurable functions be non-measurable if they are valued in a general normed space instead of $ \mathbb{R} $?

It's well known that the sum of measurable functions is measurable, if they are real or complex valued. However, the proofs I've seen heavily rely on the usage of the countable set of rational ...
5
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1answer
185 views

A name for the property $ \| x \star y \| = \| x \| \| y \| $.

Suppose that $ \star: V^2 \to V $ is some binary operation on a vector space $ V $. Should it hold, is there a name for the following property? $$ \forall x,y \in V: \quad \| x \star y \| = \| x \| \| ...
1
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1answer
62 views

Linear application on a normed space

How to prove that if $E$ and $F$ are tow normed spaces with $dim(F)<\infty$ and $f\in L(E,F)$ Then $$f~ \text{open} \Longleftrightarrow f ~\text{surjective}$$ If i suppose that $f$ is open, then ...
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1answer
56 views

Prove norm does not come from inner product.

I know I have to show it does not satisfy the parallelogram law but I don't know how to apply it.
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1answer
46 views

Test for normability of a metric on a Banach space

If d is a metric on a (finite dimensional) Banach Space and there exist norms $\|-\|_1$ and $\|-\|_2$ and constants $C_1,C_2 \in [1,\infty)$ satisfying: \begin{equation} C_1\|x-y\|_1 \leq d(x,y) \leq ...
1
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1answer
17 views

$x=argmin_{x\in A}||y-x||_2$ iff $\langle y-x,z-x\rangle \leq0$ for all $z\in A$

Consider $x\in A\subset\mathbb{R^n}$ with A closed and convex. How can you see that $$x=argmin_{x\in A}||y-x||_2$$ iff $$\langle y-x,z-x\rangle \leq0$$ for all $z\in A$. I tried using ...
0
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1answer
28 views

Estimation of the integral

I am trying to compute, or find a good estimate from above the following integral $$ \frac{1}{\pi}\int_{-\infty}^{\infty}|t|^{-1/p}\left|\frac{|t|^{\nu}-1}{t-1}\right|dt, $$where $0<1/p<1$ and ...
3
votes
5answers
59 views

Equivalence of norms: $\|x\|$ and $\|x\|_1=\|x\|+\lvert\,f(x)\rvert$

Let $f: X\to \mathbb{R}$ be a linear functional and $\|\cdot\|_1$ is defined as follow $\|x\|_1=\|x\|+|f(x)|$. Prove or disprove $\|\cdot\|_1$ is equivalent to $\|\cdot\|$ iff $f$ is continuous. I've ...
0
votes
1answer
66 views

Prob. 2, Sec. 3.3 in Erwin Kreyszig's “Introductory Functional Analysis With Applications”: How to minimise the norm?

Let $z$ be a given complex number. Let $M \subset \mathbb{C}^n$ be given by $$M \colon= \left\{ (\xi_1, \ldots, \xi_n ) \in \mathbb{C}^n \mid \sum_{i=1}^n \xi_i = z \right\}.$$ Then $M$ is convex ...