# Tagged Questions

1answer
70 views

### Is $(\ell^1 , \| \cdot \| )$ a Norm space?

Suppose $x= \{x_n \} \in \ell^ 1$ and $\| x \| = \sup | \sum_{k=1}^n x_k |$, let $\|x\|_1 = \sum_{n=1}^{\infty} |x_n |$ is a norm for $\ell^1$ . Is $(\ell^ 1 , \| \cdot \| )$ a Normed ...
0answers
35 views

### Integration for functions with values in a separable Banach space

Let $(X,\mathcal{M},\mu)$ be a measure space, $Y$ a separable Banach space, and $L_{Y}$ the space of all $(\mathcal{M},\mathcal{B}_{Y})$-measurable maps from $X$ to $Y$ (where $\mathcal{B}$ denotes ...
1answer
69 views

### How to show that this is a complete metric space [duplicate]

Let $(X;d)$ be a metric space and $\mathrm{C_b}(X,\mathbb{R})$ denote the set of all continuous bounded real valued functions defined on $X$, equipped with the uniform metric:  \rho(f,g) = \sup\{\, ...
1answer
43 views

### $L_{k}^{1}([0,1])$ is a Banach space

Let $L_{k}^{1}([0,1])$ be the space of all $f\in C^{k-1}([0,1])$ such that $f^{(k-1)}$ is absolutely continuous on $[0,1]$ (and hence $f^{(k)}$ exists a.e. and is in $L^{1}([0,1])$). Show that ...
1answer
32 views

### Condition for the unseparability of Banach Spaces

A basic question but I can't quite resolve it: Why is the following equivalent to unseparability of a Banach space X: For some uncountable set S $\subseteq$ X, there exists $\delta$ > 0 such that ...
1answer
85 views

### Completeness of $\langle \mathscr{C} [0, 1], \| \cdot \|_1 \rangle$

That's really embarrassing, however I need to ask it. I could not prove that the normed space $\langle \mathscr{C} [0, 1], \| \cdot \|_1 \rangle$ is complete (as a metric space), where \$\| f\|_1 = ...
0answers
59 views

### (Real Analysis) Integration of two functions

Note ($\Omega,A,\mu)$ is a finite additive space. Let $f\in \bar S(A, \mathbb{R})$ and $y\in Y$ where $Y$ is Banach space. Prove $\int _\Omega yf d\mu = y \int _\Omega f d\mu$. Also, for any $X\in A$, ...
3answers
82 views

### Completeness proof of $\ell^p$

Say $\{x_n\}$ is Cauchy in $\ell^p$ and $x$ is its pointwise limit. To argue that $x \in \ell^p$ would the following be correct: Let $\varepsilon > 0$ and let $N$ be s.t. $n,m > N$ ...
0answers
53 views

### A question about convergence in $L^p$. [duplicate]

Let $E$ be measurable and $1 \le p \le \infty$. Suppose $\{f_n\}_{n \in \mathbb{N}}$ all measurable and $\{f_n\}_{n \in \mathbb{N}} \to f$ pointwise a.e. $E$. For $p$ as above, I want to show that: ...
1answer
43 views

### Clarification of Reed and Simon proof of the open mapping theorem

I was reading Reed and Simon Methods of Mathematical Physics Volume 1 and have a question about a small their proof of the open mapping theorem for Banach spaces. Let $T:X\rightarrow Y$ be a bounded ...
0answers
72 views

### Show that the space $l^2= \{ a \in \mathbb{R}^\mathbb{N} | \sum |a_k|^{2} < \infty \}$ is a Banach space

If we have $l^2= \{ a \in \mathbb{R}^{\mathbb{N}} | \sum_{k=0}^{\infty} |a_k|^{2} < \infty \}$ and $||a||_2 = (\sum_{k=0}^{\infty} |a_k|^2 )^{1/2}$ 1 proposition: $l^2$ is a vector ...
2answers
127 views

### Prove that $(B, \|-\|_{\infty})$ complete. B the set of bounded real valued functions on [0,1] which are pointwise limit of continuous functions.

Question: Prove that $(B, \|-\|_{\infty})$ is complete. B the set of bounded real valued functions on [0,1] which are pointwise limit of continuous functions on [0,1]. Context: Old exam problem I'm ...
0answers
26 views

### how to show this operator is in fact convolution with a specific kernal

Firstly, the operator T is defined as taking the fourier inverse transform of the function $(1-|\zeta|)1_{[-1,1]}(\zeta)\hat{f(\zeta)}$. ie.$\hat{Tf}$=$(1-|\zeta|)1_{[-1,1]}(\zeta)\hat{f(\zeta)}$. I ...
3answers
95 views

### Does $C_0(X)$ completely determine $X$?

Let $X$ and $Y$ be compact metric spaces. Let $C_0(X)$ and $C_0(Y)$ be the Banach spaces of continuous real-valued functions over $X$ and $Y$, respectively. If $F : X \rightarrow Y$ is a ...
1answer
134 views

### Weak convergence and weak$^*$ convergence question

Let $X$ be a Banach space and $X^*$ be its dual space. Let $\phi_n\in X^\ast$ and for all $x\in X$ we have $\phi_n(x)\to c\in\mathbb{C}$ as $n\to\infty$. I want to show that the sequence $\phi_n$ has ...
1answer
96 views

### prove a subset of squence space lp closed in strong topology

Let $l^p$ be the space of $p$-summable sequences. von Neumann constructed a subset of $l^p$ space $$S=\{X_{mn}: m,n≥1\}$$ where $X_{mn}\in l^p$ are defined by $X_{mn}(m)=1, X_{mn}(n)=m$ and ...
1answer
72 views

### Proving $L^2$ convergence (application of dominated convergence?)

For any $f\in L^2(\mathbb{R}^d)$ prove \begin{align}\left\lVert \int_{\mathbb{R}^d} e^{i |x-y|^2}f(y) dy-\int_{\mathbb{R}^d}e^{i |x-y|^2} e^{-|y|^2/a}f(y) dy \right\rVert_{L^2} \rightarrow 0\ \ \ ...
0answers
46 views

### Cantor intersection theorem HW [duplicate]

Possible Duplicate: Nested sequences of balls in a Banach space I am asked to show the intersection of a decreasing sequence of nonempty closed balls in a Banach space is nonempty. It ...
2answers
199 views

### fail of cantor intersection property on closed , bounded , convex sets of integrable functions

This is from my recent homework. I am asked to find a descending nested sequence of closed , bounded , nonempty convex sets $\{D_n\}$ in $L^1(\mathbb{R})$ such that the intersection is empty , where ...
3answers
121 views

### References on relationships between different $L^p$ spaces

I am looking for detailed references containing proofs of inclusion relationships between different $L^p$ spaces and multiple counterexamples of functions in one but not the others.
1answer
123 views

### $C^1 [0,1]$ with different norm

If the space $C^1 [0,1]$ is equiped with norm $\Vert \cdot\Vert_1$,where $$\Vert f\Vert_1=|f(0)|+\Vert f'\Vert _{C}=|f(0)|+\sup_{t\in [0,1]}|f'(t)|$$ for any $f\in C^1 [0,1]$, is this space Banach? ...
1answer
96 views

### Map bounded if composition is bounded

Let $X,Y,Z$ Banach spaces and $A:X\rightarrow Y$ and $B:Y\rightarrow Z$ linear maps with $B$ bounded and injective and $BA$ bounded. Prove that $A$ is bounded as well. If I knew that $B(Y)$ is ...
2answers
185 views

### Boundary of $L^1$ space

Is there any rigorous or heuristic notion of boundary of $L^1$ that is studied? I mean something loosely like the collection of functions or distributions defined by \left\{f\notin L^1: f_n\to ...
2answers
147 views

### Shifting a function is continuous

I'm slightly puzzled by the following: if $g(t)$ is a function in $L^q(X)$ then we can show that $g(t-x)$ is continuous function of $t$, i.e. for $\varepsilon > 0$ we can find $\delta$ such that ...
1answer
353 views

### Space of Complex Measures is Banach (proof?)

How can we prove that the space of Complex Measures is Complete? with the norm of Total Variation. I have stuck on the last part of the proof where I have to prove that the limit function of a Cauchy ...
1answer
75 views

### differentiability classes

I am beginning to study multidimensional calculus using Spivak's Calculus on Manifolds, and so far I understand the purpose of considering the classes $C^n$ to be twofold. First, being in $C^n$ is a ...
1answer
114 views

### Range of bounded operator is of first category

Let $T$ be a bounded operator from a Banach Space $X$ to a normed space $Y$ such that $T$ is not onto, but $R(T)\subset Y$ is dense. Prove that $R(T)$ is of first category and not no-where dense. ...
1answer
213 views

### Every multiplicative linear functional on $\ell^{\infty}$ is the limit along an ultrafilter.

It is well-known that for any ultrafilter $\mathscr{u}$ in $\mathbb{N}$, the map\begin{equation}a\mapsto \lim_{\mathscr{u}}a\end{equation} is a multiplicative linear functional, where ...
1answer
157 views

### Cancellation law for Minkowski sums

Let $(X,\|\cdot\|)$ be a Banach space and $A,B,C\subset X$ closed bounded non-empty convex subsets. Let $+$ denote the Minkowski symbol for addition. Does the $+$ satisfy: A+C\subset B+C\implies ...
2answers
102 views

### Composition of continuous and closed operators is closed

Let $X,Y,Z$ Banach spaces, $\text{dom}(S)\subset Y$, let $T:X\rightarrow Y$ be linear and continuous and let $S:\text{dom}(S)\rightarrow Z$ be linear and closed. Show that the composition $ST$ is ...
0answers
86 views

### Question about proof of completeness of $L^p$

In my notes we prove completeness of $L^p$ by showing that if $\sum\|f_k\|_p < \infty$ then $\sum f_k$ converges in $L^p$. (That's a lemma we prove a bit earlier, namely that $(V, \|\cdot\|)$ is ...
1answer
346 views

### Inclusion of $L^p$ spaces

Let $X \subset L^1(\mathbb{R})$ a closed linear subspace satisfying \begin{align} X\subset \bigcup_{p>1} L^p(\mathbb{R})\end{align} Show that $X\subset L^{p_0}(\mathbb{R})$ for some ...
0answers
73 views

### What is the dual of the space L-infinity ($L^\infty$)? [duplicate]

Possible Duplicate: The Duals of $l^\infty$ and $L^{\infty}$ In learning real analysis, I do understand that the dual of $L^\infty$ cannot be $L^1$ because the latter is separable, whereas ...
0answers
136 views

### What is the precise definition of predual

How does one define "predual" and the surrounding notions? More specifically: Why must there be only one predual of $X$ when $X$ is a Banach space? What is the correct notion of similarity here ...
1answer
183 views

### Duality of $L^p$ and $L^q$

If $X$ is an arbitrary measure space, I already know with proof that $L^p(X)$ and $L^q(X)$ are mutually duals as Banach spaces, when $1<p$ and $p$, $q$ are dual indices. I also know a different ...
2answers
663 views

### Prove that the normed space $L^{\infty}$ equipped with $\lVert\cdot\rVert_{\infty}$ is complete. [duplicate]

Possible Duplicate: Understanding proof of completeness of $L^{\infty}$ Most of the materials I have in Real Analysis consider this statement as a trivial one: "The normed space ...
3answers
482 views

### Gâteaux derivative

Let $X$ be a Banach space and $\Omega \subset X$ be open. The functional $f$ has a Gâteaux derivative $g \in X'$ at $u \in \Omega$ if, $\forall h\in X,$ \lim_{t \rightarrow 0}[f(u+th)-f(u)- \langle ...
0answers
228 views

### A problem with $\ell_p$-norm

Let $1<p<\infty$ be fixed. Suppose $L=\{(x_1,\dots,x_n):x_i\ge 0, \sum_i x_i=1, \sum_i a_i x_i=b\}$ for some real numbers $a_i$ and $b$. I am wondering whether the following would be true. ...
3answers
100 views

### There exists a unique function $u\in C^0[-a,a]$ which satisfies this property

The problem: Let $a>0$ and let $g\in C^0([-a,a])$. Prove that there exists a unique function $u\in C^0([-a,a])$ such that $$u(x)=\frac x2u\left(\frac x2\right)+g(x),$$ for all $x\in[-a,a]$. ...
1answer
61 views

### Construct locally lipschitz map from a bounded one

Let $X$ be a Banach space and $BC(X)$ the space of all bounded closed subsets in $X$. It can be shown that $(BC(X),d_H)$ is a complete metric space (see this page for a definition of $d_H$). If ...
1answer
74 views

### Prove that the space is not complete

Let $X$ be a separable space with infinite dimension, let $(\cdot,\cdot)$ and $\|\cdot \|$ be the scalar product and the norm of $X$, and $\{e_n\}_n$ be an orthonormal basis of $X$. We define ...
0answers
45 views

### existance of the interpolation space

Let $X\subset L_1+L_2$ and let $Y$ be interpolation space between $L_2$ and $L_{\infty}.$ Given $U:X\longrightarrow Y$. My question is the following: Is there exists space $Z\subset Y$, such that ...
2answers
279 views

### Totally bounded space

Suppose $M=\left \{ f\in L^1([0,1])\, |\, 0<f(x)<\frac 1{\sqrt x} \text{almost everywhere on} \, (0,1) \right\}$. Is it true or not, that $M$ is totally bounded?
1answer
114 views

### Is this set corresponding to a bounded linear operator necessarily open?

Let $\Lambda : X \to X$ be a bounded linear operator on a Banach space $X$. My question is whether the set $$\{\lambda \in \mathbb C: \lambda I - \Lambda \quad\text{is surjective} \}$$ is ...
2answers
136 views

### How to show that the sum of $L^p$ spaces is Banach.

Let $p<q$ be positive integers (with the allowance that $q$ may be $\infty$). How can we show that the sum of $L^p$ and $L^q$ is a Banach space under the norm \$\|f\|=\inf\{\|g\|_p+\|h\|_q: ...
1answer
509 views

### If $1\leq p < \infty$ then show that $L^p([0,1])$ and $\ell_p$ are not topologically isomorphic

If $1\leq p < \infty$ then show that $L^p([0,1])$ and $\ell_p$ are not topologically isomorphic Maybe I would have to use the Rademachers.
1answer
193 views

### Swapping a limit and a $\sup$

In this proof of the completeness of $(C(K), \| \cdot \|_\infty)$ they use the following inequality:  \sup_{x \in K} \lim_{m \to \infty} | f_n(x) - f_m(x) | \leq \liminf_{m \to \infty}\, \sup_{x ...
0answers
148 views

### An explicit functional in $(l^\infty)^*$ not induced by an element of $l^1$? [duplicate]

Possible Duplicate: Nonnegative linear functionals over $l^\infty$ Setup: Let $l^\infty$ be the set of bounded sequences (with terms in $\mathbb{R}$), and let $l^1$ be the set of sequences ...
4answers
301 views

### Does the completeness of a normed vector space only depend on its topology?

Let $V \space$ be a vector space over $\mathbb{R}$, and $\Vert \cdot \Vert_1$, $\Vert \cdot \Vert_2$ norms over $V$, which generate the same topology. Is it always true that if $v_n$ is a Cauchy ...
1answer
183 views

### Cauchy, Bolzano-Weierstrass, Convergence

Why is it that if for every bounded sequence we can find a convergent subsequence (in a normed vector space) then every Cauchy sequence converges (in this normed space)? Thanks.
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