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

19
votes
1answer
6k views

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?
16
votes
2answers
7k views

“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 ...
5
votes
2answers
4k views

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 ...
16
votes
1answer
5k views

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 ...
5
votes
1answer
2k views

$C([0, 1])$ 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 ...
5
votes
2answers
464 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
votes
1answer
1k views

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 ...
4
votes
1answer
3k views

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 ...
33
votes
5answers
7k views

Not every metric is induced from a norm

I have studied that every normed space $(V, \lVert\cdot \lVert)$ is a metric space with respect to distance function $d(u,v) = \lVert u - v \rVert$, $u,v \in V$. My question is whether every metric ...
12
votes
3answers
4k views

Every proper subspace of a normed vector space has empty interior

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
votes
3answers
2k views

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 ...
53
votes
3answers
2k views

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 ...
20
votes
2answers
7k views

Why is $L^{\infty}$ not separable?

$l^p (1≤p<{\infty})$ 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 ...
12
votes
2answers
2k views

$T$ 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 ...
7
votes
1answer
2k 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.
3
votes
2answers
697 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 ...
3
votes
1answer
431 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 ...
1
vote
1answer
229 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 ...
36
votes
2answers
10k views

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 ...
8
votes
1answer
2k views

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 ...
12
votes
2answers
6k views

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 ...
5
votes
1answer
187 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 \| \| ...
7
votes
5answers
1k views

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.
6
votes
2answers
3k views

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 ...
2
votes
1answer
782 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 ...
3
votes
1answer
770 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.
0
votes
1answer
413 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 $x$ ...
25
votes
3answers
2k views

Why are every structures I study based on Real number?

I've been studying basic concepts of inner product vector space, normed vector space and metric space. And all the inner products, norms and metrics are defined to be real-valued functions in my ...
9
votes
3answers
614 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?
9
votes
2answers
128 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 ...
4
votes
2answers
272 views

Hahn-Banach Theorem for separable spaces without Zorn's Lemma

I was reading about the Hahn-Banach Theorem, its many versions and their proofs. It's known that in the proofs we need Zorn's Lemma. But in the book that I'm reading, the author said if $X$ is a ...
3
votes
1answer
743 views

Banach spaces and their unit sphere

Let $X$ be a normed vector space. Show that if a subsequence of a Cauchy sequence converges, then the whole sequence converges. Use the part 1 to show that $S = \{x\in X : \|x\| = 1\}$ is complete ...
11
votes
2answers
992 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 ...
8
votes
1answer
2k views

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 ...
5
votes
1answer
2k views

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 ...
3
votes
2answers
152 views

TVS: Topology vs. Scalar Product

I just had an idea: It is clear that every scalar product induces a norm and that a metric and that finally a topology. Turning this argument around we know: Not every topology induces a metric only ...
3
votes
1answer
779 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 $d(x,Y)>1-\epsilon$. ...
5
votes
1answer
473 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: ...
5
votes
2answers
986 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 ...
1
vote
1answer
103 views

A Cauchy sequence has a rapidly Cauchy subsequence

I am trying to fill in the details of a proof related to the Riesz-Fischer Theorem. We need to show that every Cauchy sequence $\{f_n\}$ has a rapidly Cauchy subsequence. My text claims that we can ...
0
votes
2answers
493 views

Convex set weakly closed if and only if strongly closed as well

I'm looking for a proof that given $(X\textbf{ } \|\cdot\|)$ normed space, $M \subset X$ convex set, $M$ is weakly closed if and only if it's strongly closed as well.
6
votes
3answers
3k views

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 ...
3
votes
1answer
108 views

Every Banach space is isomorphic to $\ell_1/A$ for some closed $A\subset \ell_1$

How to prove the following mind-blowing fact? Let $(X, \|\ \|)$ be a separable Banach space, $\ell_1\subset \mathbb{R}^\infty$ - the space of absolutely summable scalar sequences. Then there ...
3
votes
3answers
876 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 ...
8
votes
2answers
2k views

Examples of compact sets that are infinite dimensional and not bounded

In an infinite dimensional Banach space, does a compact subset have to be finite dimensional? I know it cannot contain any infinite dimensional balls, if this mean it has to be finite dimensional, ...
16
votes
1answer
11k views

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 appears that in $R^{n}$ a number of ...
13
votes
3answers
2k views

Does every $\mathbb{R},\mathbb{C}$ vector space have a norm?

Is there a canonical way to define on any vector space over $\mathbb{K}=\mathbb{R},\mathbb{C}$ a norm ? (Or, if there isn't, can someone give me an example of a vector space over $\mathbb{K}$ that is ...
39
votes
1answer
1k views

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 ...
10
votes
1answer
2k views

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 ...
6
votes
1answer
3k views

About Banach Spaces And Absolute Convergence Of Series

How to prove the following two assertions: If in a normed space $X$, absolute convergence of any series always implies convergence of that series, then $X$ is a Banach space. In a Banach space, ...