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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Let $X$ be an infinite dimensional Banach space. Prove that every Hamel basis of X is uncountable.

Let $X$ be an infinite dimensional Banach space. Prove that every basis of $X$ is uncountable. Can anyone help how can I solve the above problem?
3
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2answers
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Prove that $X'$ is a Banach space

I'm taking a new course on functional analysis and meet with the following problem. If $X$ is a normed space (not necessarily complete), then prove that $X'$ is a Banach space. Definition: When the ...
5
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2answers
393 views

If $ x_n \to x $ then $ z_n = \frac{x_1 + \dots +x_n}{n} \to x $

I have this problem I'm working on. Hints are much appreciated (I don't want complete proof): In a normed vector space, if $ x_n \longrightarrow x $ then $ z_n = \frac{x_1 + \dots +x_n}{n} ...
14
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1answer
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Open Mapping Theorem: counterexample

The Open Mapping Theorem says that a linear continuous surjection between Banach spaces is an open mapping. I am writing some lecture notes on the Open Mapping Theorem. I guess it would be nice to ...
51
votes
3answers
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Explicit norm on $\mathcal{C}^0(\mathbb{R},\mathbb{R})$

Do you know an explicit norm on $\mathcal{C}^0(\mathbb{R},\mathbb{R})$? Using the axiom of choice, every vector space admits a norm but have you an explicit formula on ...
12
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2answers
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“Every linear mapping on a finite dimensional space is continuous”

From Wiki Every linear function on a finite-dimensional space is continuous. I was wondering what the domain and codomain of such linear function are? Are they any two topological vector ...
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2answers
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Why are $l^{\infty}$ and $L^{\infty}$ non separable spaces?

$l^p (1≤p<{\infty})$ and $L^p (1≤p<∞)$ are separable spaces. What on earth has changed when the value of p turns from a finite number to ${\infty}$? Our teacher gave us some hints that there ...
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1answer
203 views

Prove that the closed unit ball of $L^2[a,b]$ is closed in $L^1[a,b]$

Prove that the closed unit ball of $L^2[a,b]$ is closed in $L^1[a,b]$. My friends and I have literally been pouring over this problem for days now without success. We've been using Hölder's ...
26
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2answers
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Difference between metric and norm made concrete: The case of Euclid

This is a follow-up question on this one. The answers to my questions made things a lot clearer to me (Thank you for that!), yet there is some point that still bothers me. This time I am making ...
7
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1answer
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Are isometric normed linear spaces isomorphic?

I should know the answer to this (and I did some time ago, but have forgotten): If the normed linear spaces $X$ and $Y$ are isometric (there is a bijective map from $X$ to $Y$ that preserves ...
7
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1answer
970 views

Equivalence of reflexive and weakly compact

In a normed space $X$ is there an equivalence between these two proposition? $1)$ $X$ is reflexive; $2)$ $B$, the unit ball of $X$, is weakly compact.
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equivalent norms in Banach spaces of infinite dimension

Suppose $ X $ is a Banach space with respect to two different norms, $ \|\cdot\|_1 \mathrm{ e } \|\cdot\|_2 $. Suppose there is a constant $ K > 0 $ such that $$ \forall x \in X, \|x\|_1 \leq ...
2
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1answer
538 views

The vector space formed on C[0,1] and the norm $(\int_{0}^{1}|f(t)|^2 dt)^{1/2} $

How does one show that $(C[0,1], ||.||_{2})$ where $||.||_2=(\int_{0}^{1}|f(t)|^2dt)^{1/2}$ and $C[0,1]$ is the space of all continous function which are mapped from $[0,1]\rightarrow \mathbb{R}$, is ...
4
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2answers
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How to prove that if a normed space has Schauder basis, then it is separable? What about the converse?

Can we take as a dense subset the collection of all the linear combinations of the vectors of the Schauder basis using the rationals as scalars (or the complex numbers with rational real and imaginary ...
3
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2answers
447 views

The weak$^*$ topology on $X^*$ is not first countable if $X$ has uncountable dimension.

I learnt without proof that if $X$ is a normed space of uncountable dimension, then the weak* topology on $X^*$ is not first countable. Can anyone point out how I should go about proving it? I tried ...
2
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1answer
260 views

If the dual spaces are isometrically isomorphic are the spaces isomorphic?

Let $X$, $Y$ be Banach spaces such that the duals $X^\ast$ and $Y^\ast$ are isometrically isomorphic. Are $X$ and $Y$ necessarily isomorphic? The answer to the question whether $X$ and $Y$ are ...
9
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3answers
521 views

Is there an easy example of a vector space which can not be endowed with the structure of a Banach space

Let $V$ be a real vector space. Is there always a norm on $V$ such that $V$ is complete with respect to this norm? If not, is there an easy counterexample?
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$T$ is continuous if and only if $\ker T$ is closed

Let $X,Y$ be normed linear spaces. Let $T: X\to Y$ be linear. If $X$ is finite dimensional, show that $T$ is continuous. If $Y$ is finite dimensional, show that $T$ is continuous if and only if $\ker ...
11
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2answers
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Is a norm a continuous function?

Is a norm on a set a continuous function with respect to the topology induced by the norm? Is a topology on the set that can make the norm continuous (i.e. the topology that is compatible with the ...
11
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2answers
684 views

How to show that $\mathbb R^n$ with the $1$-norm is not isometric to $\mathbb R^n$ with the infinity norm for $n>2$?

Could you please give me a hint to prove that $\mathbb{R}^n$ with the 1-norm $\lvert x\rvert_1=\lvert x_1\rvert+\cdots+\lvert x_n\rvert$ is not isometric to $\mathbb{R}^n$ with the infinity-norm ...
7
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3answers
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Interior of a Subspace

There is a conjecture: "The only subspace of a normed vector space $V$ that has a non-empty interior, is $V$ itself." (here, the topology is the obvious set of all open sets generated by the metric ...
4
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1answer
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Show that the norm of the multiplication operator $M_f$ on $L^2[0,1]$ is $\|f\|_\infty$

I'm having some (hopefully small) issues computing the norm of an operator. Firstly, the problem, For $f\in L^\infty[0,1]$, define $M_f: L^2[0,1]\to L^2[0,1]$ by $M_f(g)(x) = f(x)g(x)$. Show that ...
5
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5answers
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Fixed point Exercise on a compact set

Let $K$ a compact normed space and $f:K\rightarrow K$ so that $$\|f(x)-f(y)\|<\|x-y\|\quad\quad\forall\,\, x, y\in K, x\neq y.$$ Prove that $f$ have a fixed point.
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2answers
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Subspaces of separable normed spaces

Let $X$ be a separable normed space. Is it true that every subspace is separable? If it was Hilbert space I would take the dense set and then their projections. It sounds trivial but I cannot prove ...
2
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1answer
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Finite-dimensional subspace normed vector space is closed

I know that probably this question has already been answered, but I'd like to present my attempt of solution. Let $(E,\|\cdot\|)$ be a normed vector space and let $F\subseteq E$ be a ...
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1answer
61 views

Orthogonal in inner product space

Let $(X,<.>)$ is an inner product space prove that $x$ and $y$ are orthogonal if and only if $||x+αy|| \ge ||x||$ for any scalar $α$ . The first direction if $x$ and $y$ are orthogonal ...
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2answers
58 views

Some question about extension of bounded linear operator

Let we have the following bounded linear operator $$T: D(T)\rightarrow Y$$ such that $D(T)$ is the domain and it is a vector space and $Y$ is a Banach space . Then it has an extension $$H: ...
3
votes
1answer
544 views

If $X^\ast $ is separable $\Longrightarrow$ $S_{X^\ast}$ is also separable

Let $X$ be a Banach space such that $X$* (Dual space of $X$) is separable How can we prove that $S_{X^\ast}$ (Unit sphere of $X$*) is also separable Any hints would be appreciated.
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1answer
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Banach Spaces - How can $B,B',B'', B''', B'''',B''''',\ldots$ behave?

(ZFC) Let $ \big\langle B,+,\cdot, \:\: \|\cdot\| \:\: \big\rangle $ be a Banach space. Define $ \mathbf{B} \; = \;\big\langle B,+,\cdot, \:\: \|\cdot\| \:\: \big\rangle $. Define $\: \mathbf{B}_0 ...
9
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1answer
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Relations between p norms

The p-norm is given by $||x||_{p} = (\sum_{n=1}^{\infty }|x_{n}|^{p})^{1/p}$. For $0 < p < q$, it can be shown that $||x||_{p} \geq ||x||_{q}$ (1, 2). It sppears that in $R^{n}$ a number of ...
3
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1answer
515 views

A stronger statement of Riesz's lemma

Riesz's lemma state that If $Y$ is a proper, closed subspace of a normed space $X$, then for any $\epsilon>0$, there exists $x$ in the closed unit ball of $X$ such that $d(x,Y)>1-\epsilon$. ...
2
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1answer
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$C([0, 1])$ is not complete space with respect to norm $\lVert f\rVert _1 = \int_0^1 \lvert f (x) \rvert \,dx$

Consider $C([0, 1])$, the linear space of continuous complex-valued functions on the interval $[0, 1]$, with the norm $\displaystyle\lVert f\rVert_1 = \int_0^1 \lvert f(x)\rvert \,dx$. I have to ...
7
votes
1answer
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An example of a norm which can't be generated by an inner product

I realize that every inner product defined on a vector space can give rise to a norm on that space. The converse, apparently is not true. I'd like an example of a norm which no inner product can ...
5
votes
2answers
667 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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Proof that every finite dimensional normed vector space is complete

Can you read my proof and tell me if it's correct? Thanks. Let $V$ be a vector space over a complete topological field say $\mathbb R$ (or $\mathbb C$) with $\dim(V) = n$, base $e_i$ and norm ...
2
votes
1answer
341 views

continuous projections to finite dimensional subspaces of normed spaces

If $X$ is a normed space and $Y$ is a finite dimensional subspace, then there exists a continuous linear projection $P$ from $X$ to $Y$. Our teacher gave us the instruction to use the following fact: ...
2
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3answers
967 views

Cauchy-Schwarz Inequality

In Luenberger book Cauchy-Schwarz Inequality is defined like this: For all $x,y$ in an inner product space $|(x|y)| \le \|x\|\|y\|$. Equality holds if and only if $x = \lambda y$ or $y = \theta$. ...
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1answer
84 views

Proving that $AB-BA=cI$ for nontrivial $c \in \mathbb{C}$

I have a homework question I can`t solve: Let $X$ be a normed linear space, $A,B \in B(X)$. Show that there exists no nontrivial $c \in \mathbb{C} $ such that $AB-BA=cI$. Thanks alot already guys! I ...
0
votes
2answers
227 views

Norm equivalence (Frobenius and infinity)

I am trying to prove the matrix norm equivalence for norms 1, 2, $\infty$ and Frobenius. I have managed to solve find the constants for $||.||_{1}$ and $||.||_{2}$ but I cannot see how to continue ...
0
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1answer
105 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 ?
0
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1answer
527 views

Operator Norm of a Linear Transformation

PROBLEM For the linear transformation $T:\mathbb{R}^n\rightarrow\mathbb{R}^m$ equipped with the $l^1$-norm, namely, for $x\in\mathbb{R}^n$, $||x||=\sum_{j=1}^n |x_j|$ and similarly for ...
5
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2answers
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Derivation of the polarization identities?

For a real (or complex) inner product space $V$, the inner product can be expressed in terms of the norm as either $$ \langle x,y\rangle=\frac{1}{4}(\|x+y\|^2-\|x-y\|^2) $$ or $$ \langle ...
4
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1answer
256 views

Shortest path on unit sphere under $\|\cdot\|_\infty$

Let $X$ be $\mathbb{R}^3$ with the sup norm $\|\cdot\|_{\infty}$. Let $Y=\{x\in X: \|x\|_{\infty}=1\}$. For $x,y\in Y$ define $d(x,y)$ to be the arc length of shortest paths on $Y$ joining $x,y$. (It ...
4
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1answer
260 views

Non-completeness of the space of bounded linear operators

If $X$ and $Y$ are normed spaces I know that the space $B(X,Y)$ of bounded linear functions from $X$ to $Y$, is complete if $Y$ is complete. Is there an example of a pair of normed spaces $X,Y$ s.t. ...
3
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1answer
236 views

convergence of sequence of averages the other way arround

In a vector normed space, if $ \{x_n\} \longrightarrow x $ then $ z_n = \dfrac{x_1 + \cdots+x_n}{n} \longrightarrow x $ Is it true the other way arround too? meaning: if $ z_n = \dfrac{x_1 + ...
2
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2answers
420 views

Any two norms on finite dimensional space are equivalent

Any two norms on a finite dimensional linear space are equivalent. Suppose not, and that $||\cdot||$ is a norm such that for any other norm $||\cdot||'$ and any constant $C$, $C||x||'<||x||$ for ...
2
votes
3answers
488 views

Boundedness and pointwise convergence imply weak convergence in $\ell^p$

Let $p\in(1,+\infty)$ and consider the space $\ell^p$ with its usual norm. The following are equivalent: (1) $x_n \rightharpoonup x$ (i.e. $x_n$ weakly converges to $x$); (2) $$\exists ...
1
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1answer
43 views

Computing $\sup_{\left \| u \right \|=1} d(u,F)$ in a closed subspace of a normed space.

I have come across with the following problem: Let $E$ be a normed space and $F$ a closed subspace of $E$. It's asking to compute $\sup\limits_{\left \| u \right \|=1} d(u,F)$. What I know it's ...
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vote
1answer
74 views

Can $||f|| = ||a||_{q}$ to arbitrary values of $p$ and $q$ satisfying ${1 \over p} + {1 \over q} = 1$

We all know that: Suppose $a = (a_{1}, a_{2}, ..., a_{n})$ is a point in Euclide space $R^{n}$. Consider the mapping $f: R^{n} \rightarrow R$, $f(x) = \sum_{i=1}^{n}a_{i}x_{i}$. Then $||f|| = ...
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vote
2answers
142 views

Characterisation of norm convergence

Let $X$ be a Hilbert space and $(x_n)\in X^{\mathbb N}$ be a sequence. Then the following statements are equivalent (with $x \in X$): We have $x_n \to x$, i.e. $\| x_n -x \| \to 0$ and we have $x_n ...