Functional analysis is the study of infinite-dimensional vector spaces, often with additional structures (inner product, norm, topology), with typical examples given by function spaces. The subject also includes the study of linear and non-linear operators on these spaces, as well as measure, ...

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7
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406 views

On the weak and strong convergence of an iterative sequence

I have some difficulties in the following problem. I would like to thank for all kind help and construction. Let $H$ be an infinite dimensional real Hilbert space and $F: H\rightarrow H$ be a ...
6
votes
1answer
1k views

Finite-dimensional subspaces of normed vector spaces are direct summands

Here is a problem in functional analysis from Folland's book: If $\mathcal{M}$ is a finite-dimensional subspace of a normed vector space $\mathcal{X}$, then there is a closed subspace ...
5
votes
1answer
225 views

The sup norm on $C[0,1]$ is not equivalent to another one, induced by some inner product

Let $\mathrm{C}[0,1]$ be the space of continuous functions $[0,1]\rightarrow \mathbb{R}$ endowed with the norm $||x||_{\infty}=\mathrm{max}_{t\in [0,1]}|x(t)|$. It is easy to verify that this norm is ...
5
votes
5answers
603 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.
5
votes
2answers
315 views

If $(I-T)^{-1}$ exists, can it always be written in a series representation?

If $X$ is a Banach space, and $T:X \to X$ is a bounded linear operator with norm < $1$, then $I-T$ has a bounded inverse defined by $(I-T)^{-1} = \sum_{n=0}^\infty T^n$. Thinking in terms of a ...
4
votes
1answer
116 views

Continuous function on closed unit ball

Take a continuous mapping $f: \bar{B^{n}} \rightarrow \bar{B^{n}}$, where $\bar{B^{n}}$ is a closed unit ball in $\mathbb{R}^{n}$. Assume that $f(x) \neq x$ for every $x \in \bar{B^{n}}$. Define ...
4
votes
1answer
286 views

Lower Semicontinuity Concepts

Let $X$ be a real Banach space, let $f:X\rightarrow \overline{\mathbb{R}}$ be a functional. We have known that: If $f$ is weakly lower semicontinuous then $f$ is weakly sequentially lower ...
4
votes
2answers
204 views

Can all real/complex vector spaces be equipped with a Hilbert space structure?

Let $X$ be a vector space over $\mathbb K \in \{\mathbb R, \mathbb C\}$. Does there exists a pairing $X \times X \rightarrow \mathbb K$ that induces a Hilbert space structure on $X$? I have been ...
3
votes
2answers
818 views

Is a von Neumann algebra just a C*-algebra which is generated by its projections?

von Neumann algebras have the nice property that they are generated by their projections (the elements satisfying $e = e^{\ast} = e^2$) in the sense that they are the norm closure of the subspace ...
2
votes
1answer
1k views

Proving that the dual of the $\mathcal{l}_p$ norm is the $\mathcal{l}_q$ norm.

Let $\| \cdot \|$ be a norm on $\mathbb{R}^n$. The associated dual norm, denoted $\| \cdot \|_*$ is defined as $\| z \|_* = \sup\{ z^{t} x : \| x \| < 1 \}$. Does someone know how prove that the ...
1
vote
1answer
585 views

Show that $\liminf_{n\to \infty}x_{n}\le\alpha(x)\le\limsup_{n\to\infty}x_{n}$ for $x=(x_{n})$ in $\ell^{\infty}$

Question: Show that $\liminf_{n\to \infty}x_{n}\le\alpha(x)\le\limsup_{n\to\infty}x_{n}$ for $x=(x_{n})$ in $\ell^{\infty}$, where $\alpha$ is a bounded linear functional on $\ell^{\infty}$. I ...
13
votes
1answer
743 views

Different versions of Riesz Theorems

In Wikipedia, there are three versions of Riesz theorems: 1 The Hilbert space representation theorem for the (continuous) dual space of a Hilbert space; 2 The representation theorem for ...
11
votes
2answers
457 views

Is the closedness of the image of a Fredholm operator implied by the finiteness of the codimension of its image?

Let $X$ and $Y$ be Banach spaces. A bounded operator $T\colon X\to Y$ is called Fredholm iff The dimension of $\ker(T)$ is finite, The codimension of the image $\mathrm{im}(T)$ is finite, The image ...
7
votes
1answer
618 views

Urysohn's function on a metric space

Let $(X,d)$ be a metric space and $A\subset B\subset X$. $A$ is closed, $B$ is open. If there are developed methods to find at least one (or describe the whole class) of Urysohn's functions for $A$ ...
7
votes
1answer
673 views

Do there exist closed subspaces $X$, $Y$ of Banach space, such that $X+Y$ is not closed?

I am looking for an example of two closed subspaces of a Banach space, such that their sum is not closed.
6
votes
2answers
977 views

On the weak closure

Let $\lbrace e_n \rbrace$ for the standard unit vectors in $l_2$. I want to show that $0$ is in weak closure of $\lbrace\sqrt{n}e_n\rbrace$ but no subsequence of $\lbrace \sqrt{n}e_n\rbrace$ weakly ...
6
votes
1answer
516 views

Dual of $\ell_{\infty}$ is not $\ell_1$

As the title indicates I'm trying to show that $\ell_{\infty}^{*}$ is not $\ell_1$. I've shown that for p, q conjugate and finite we do indeed have $\ell_{p}^{*} = \ell_q$, with the correspondence ...
5
votes
1answer
774 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.
5
votes
1answer
161 views

Why $L^{r}(X)\cap L^{t}(X)\subset L^{s}(X)$ for $1<r<s<t$?

I am working on this homework problem, and I am totally stuck: Let $(X,\mu)$ be a measure space, and let $1 \leq r < s < t < \infty$. Prove that there exist constants $\alpha,\beta>0$ so ...
4
votes
1answer
381 views

Counterexample for the Open Mapping Theorem

I would like to ask a counterexample for the open mapping theorem: Find a discontinuous linear mapping $T:X \to Y$ such that $T(X)=Y$ and $X,\;Y$ are Banach but $T$ is not open. Could you help me ...
3
votes
1answer
181 views

When is a subset of $\ell^2$ compact?

I have been looking on the internet for hours now and even asking in chat without an answer. When is a set $M\subseteq\ell^2$ compact? For $L^p$, there is the Arzelà–Ascoli theorem that provides a ...
3
votes
1answer
183 views

Hilbert's Inequality

Could you help me to show the following: The operator $$ T(f)(x) = \int _0^\infty \frac{f(y)}{x+y}dy $$ satisfies $$\Vert T(f)\Vert_p \leq C_p \Vert f\Vert_p $$ for $1 <p< \infty$ where ...
3
votes
2answers
558 views

Hellinger-Toeplitz theorem use principle of uniform boundedness

Suppose $T$ is an everywhere defined linear map from a Hilbert space $\mathcal{H}$ to itself. Suppose $T$ is also symmetric so that $\langle Tx,y\rangle=\langle x,Ty\rangle$ for all ...
3
votes
1answer
225 views

weak$^*$-separability of $l_\infty^*$.

Where can I find the proof that $l_\infty^*$ is weeak$^*$-separable? I want to re-examine the proof of that fact.
2
votes
1answer
125 views

Relation between Sobolev Space $W^{1,\infty}$ and the Lipschitz class

I have a Sobolev space related question. In the book 'Measure theory and fine properties of functions' by Lawrence Evans. I know the result that states that for $f: \Omega \rightarrow \mathbb{R}$. $f$ ...
2
votes
1answer
974 views

Is any closed ball non-compact in infinite dimensional space?

It is known that the closed unit ball $\overline{B_1(0)}$ in a normed space $X$ is compact if and only if $\dim X < \infty$. In particular, the $\overline{B_1(0)}$ is not compact if $\dim X = ...
2
votes
4answers
624 views

Space of bounded functions is reflexive if the domain is finite

Let $C_b(X)$ be a space of bounded continuous functions on a locally compact space $X$ equipped with the supremum norm. How to show that $C_b(X)$ is reflexive if and only if $X$ is finite?
2
votes
2answers
296 views

First theorem in Topological vector spaces.

I came across this theorem and I am disappointed not being able to understand or to have intuition to understand it . I would be glad to get help . Theorem : If $K$ and $C$ are subset of topological ...
0
votes
1answer
238 views

Nested sequence of sets in Hilbert space [duplicate]

How can I prove that nested sequence of non-empty bounded closed convex sets in Hilbert space have nonempty intersection? I just don't know where to start. Thanks
8
votes
1answer
568 views

A net version of dominated convergence?

Let $G$ be a locally compact Hausdorff Abelian topological group. Let $\mu$ be a Haar measure on $G$, i.e. a regular translation invariant measure. Let $f$ be fixed in $\mathcal{L}^1(G, \mu)$. ...
7
votes
2answers
621 views

Linear functional on a Banach space is discontinuous then its nullspace is dense.

I need to prove that: If a nonzero linear functional $f$ on a Banach Space $X$ is discontinuous then the nullspace $N_f$ is dense in $X$. To prove that $N_f$ is dense, it suffices to show that ...
5
votes
2answers
117 views

Differentiable $f$ such that the set of translates of multiples of $f$ is a vector space of dimension two

How can we derive all of the differentiable functions $f \colon \mathbb{R} \to \mathbb{R}$ such that $V=\{af_b : a,b \in \mathbb{R}\}$ is a vector space of dimension two, where $f_b\colon \mathbb{R} ...
5
votes
1answer
310 views

Question about definition of Sobolev spaces

I'm trying to understand the following definition: which can also be found here on page 136. Question 1: Closure with respect to what norm? It's not given in the definition. Question 2: Do I have ...
4
votes
2answers
263 views

Banach-Stone Theorem

In here, Banach first proved a lemma which used directional derivative to identity peak point of functions. Then he used the lemma in the proof of Banach-Stone theorem. After several years, Stone ...
4
votes
1answer
166 views

Prove $\ell_1$ is first category in $\ell_2$

Prove that $\ell_1$ is first category in $\ell_2$. I tried to solve this, but had no idea about the approach. Any suggestions are helpful. Thanks in advance.
4
votes
1answer
152 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 ...
4
votes
2answers
626 views

The Space $C(\Omega,\mathbb{R})$ has a Predual?

Let $\Omega$ be a compact metric space and $C(\Omega,\mathbb{R})$ the space of borel continuous real valued functions. I would like to know if there is any real Banach space $V$ such that its dual ...
4
votes
1answer
361 views

$\ell^1$ vs. continuous dual of $\ell^{\infty}$ in ZF+AD

Let the base field be the real numbers or the complex numbers (I don't think it will matter). Let $(\ell^{\infty})'$ be the continuous dual of the Banach space $\ell^{\infty}$. Let $\: f : \ell^1 ...
4
votes
1answer
252 views

Why is $GL(B)$ a Banach Lie Group?

Banach Lie Groups are what you'd expect: http://www.encyclopediaofmath.org/index.php/Lie_group,_Banach If $B$ is a Banach algebra then why is $GL(B)$, the set of invertible elements of $B$, a ...
4
votes
1answer
706 views

Weak convergences of measurable functions and of measures

My question is "how weak convergences of measurable functions is defined?" There seems to be two different definitions which are both based on weak convergence of measures generated by the measurable ...
3
votes
4answers
163 views

Topological Vector Space: $\dim Z\text{ finite}\implies Z\text{ closed}$

Let $V$ be a Hausdorff topological vector space and $Z$ a linear subspace: $Z\leq X$ Is there a neat way to prove that: $$\dim Z\text{ finite}\implies Z\text{ closed}$$
3
votes
1answer
415 views

Given a point $x$ and a closed subspace $Y$ of a normed space, must the distance from $x$ to $Y$ be achieved by some $y\in Y$?

I think no. And I am looking for examples. I would like a sequence $y_n$ in $Y$ such that $||y_n-x||\rightarrow d(x,Y)$ while $y_n$ do not converge. Can anyone give a proof or an counterexample to ...
3
votes
3answers
139 views

If for every $v\in V$ $\langle v,v\rangle_{1} = \langle v,v \rangle_{2}$ then $\langle\cdot,\cdot \rangle_{1} = \langle\cdot,\cdot \rangle_{2}$

Let $V$ be a vector space with a finite Dimension above $\mathbb{C}$ or $\mathbb{R}$. How does one prove that if $\langle\cdot,\cdot\rangle_{1}$ and $\langle \cdot, \cdot \rangle_{2}$ are two Inner ...
2
votes
1answer
856 views

Why is $l^\infty$ not separable?

My functional analysis textbook says "The metric space $l^\infty$ is not separable." The metric defined between two sequences $\{a_1,a_2,a_3\dots\}$ and $\{b_1,b_2,b_3,\dots\}$ is ...
2
votes
1answer
87 views

Find the weak sequential closure of a set in $L^2(-\pi,\pi)$

$A=\{f_{m,n}(t)|0\le m<n\}$ where $f_{m,n}(t)=e^{imt}+me^{int}$. I should find the weak sequential closure of $A\subset L^2(-\pi,\pi)$. I know what I'm supposed to do. Take a sequence in $A$ and ...
2
votes
1answer
78 views

Question about SOT and compact operators

I need some help with functional analysis / Hilbert space theory. If you have a favorite text to recommend, please let me know~ Here is my question: Given $v_t$ be the "squeeze operator" on ...
2
votes
2answers
214 views

variational problem-exercice

Let $\Omega$ an open bounded connexe and regular, and let $f \in L^2(\Omega)$ We consider the variational problem: find $u \in H^1(\Omega)$ such: $$\displaystyle\int_{\Omega} A \nabla u \cdot \nabla v ...
2
votes
0answers
367 views

Why in uniformly convex Banach space every non empty, closed, convex subset contains a unique element of smallest norm?

In Hilbert space every non empty, closed, convex subset contains a unique element of smallest norm. Why is that true also in Banach space which is uniformly convex? (normed space which is uniformly ...
2
votes
2answers
252 views

Openness of linear mapping

Let $X$ be a topological vector space over the field $K$, where $K=\mathbb R$ or $K= \mathbb C$, and let $f:X\rightarrow K^n$ ($n \in \mathbb N$) be a linear and surjective functional. How to prove ...
1
vote
4answers
161 views

Prove that some topology is not metrizable

Suppose I am asked to show that some topology is not metrizable. What I have to prove exactly for that ?