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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3
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1answer
177 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
674 views

Every Hilbert space has an orthonomal basis - using Zorn's Lemma

The problem is to prove that every Hilbert space has a orthonormal basis. We are given Zorn's Lemma, which is taken as an axiom of set theory: Lemma If X is a nonempty partially ordered set with the ...
3
votes
2answers
486 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 ...
2
votes
2answers
114 views

Resolvent Set: Definition

Given Banach spaces: $X,Y$ Consider a linear operator: $T:\mathcal{D}(T)\to Y$ (not necessarily bounded nor closed nor closable nor densely defined) Define for the shorthand the shifted operator: ...
2
votes
1answer
189 views

Help proving that a Banach space is reflexive

I'm having trouble in proving that the following space is reflexive: $$E = \{ x= (x_n) : x_n \in \mathbb{R}^n \text{ and } \sum \|x_n\|^2_\infty < \infty\}$$ with the norm $$ \|x\| = (\sum ...
2
votes
4answers
550 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
282 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 ...
1
vote
1answer
168 views

Show $\{u_n\}$ orthonormal, A compact implies $\|Au_n\| \to 0$

I'm having a bit trouble with this homework exercise. Let $\mathcal{H}$ be a Hilbert space and $\{u_n\}_{n=1}^\infty$ an orthonormal sequence in $\mathcal{H}$. Let $A$ be a compact operator on ...
0
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1answer
222 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
7
votes
1answer
512 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
537 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 ...
7
votes
1answer
200 views

$L_{p}$ distance between a function and its translation

I'm working through a proof and one of the comments is that for a function $f\in L_p (\mathbb{T})$: $$\lim_{t\to \infty}\;\|f(\cdot + t) - f\|_p = 0.$$ Should this read as $t\to 0$? If so, how do ...
6
votes
2answers
914 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 ...
5
votes
2answers
116 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
275 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
220 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
153 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
147 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
573 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
341 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
227 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
634 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
151 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
3answers
135 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
259 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
81 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
72 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
213 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
1answer
326 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 ...
2
votes
0answers
352 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 ...
1
vote
1answer
75 views

Any idea with this problem of distances???

Let $E$ a normed linear space and $H$ the closed hyperplane $H=\ker f$, where $f\in L(E,\mathbb{R})$, $f\not\equiv 0$. Show that if $a\in E$ then $$d(a,H)=\frac{|f(a)|}{||f||}$$ And the problem have a ...
1
vote
2answers
381 views

how to show that the norm-limit of compact operators is compact?

When I read the item of compact operator on Wikipedia, it said that Let $T_{n}, n\in \mathbb{N}$, be a sequence of compact operators from one Banach space to the other, and suppose that $Tn$ ...
1
vote
1answer
77 views

Does this sequence of operators in Hilbert space, given by an algorithm, terminate

Let $H$ be an infinitedimensional Hilbert space and $T$ a compact selfadjoint operator in it. Consider the following Algorithm: Let $$ H_{1}=H,\ T_{1}=T $$ and let $\lambda_{1}$ be that ...
0
votes
1answer
68 views

prove that the operator is compact.

Let $H$ be a Hilbert space over $\mathbb C$, and $\{f_j\}$ a orthonormal set in $H$. Let $t_j\in \mathbb C$ such that $\displaystyle \lim_{n\to \infty} t_j =0$ i.e $(t_j)_{j\in \mathbb N}\in c_0$. ...
-2
votes
1answer
145 views

Let $f$ be a function $f:\Bbb R\to \Bbb R$ and $f^{−1} (F)$ for every $F$ subset of $\Bbb R$ and finite $F$ be measurable which is correct?

Let $f$ be a function $f: \mathbb{R} \to \mathbb{R}$ . $|f|$ is measurable and $f^{−1} (F)$ for every $F\subset \mathbb{R}$ is measurable too (and $F$ is finite) which of the following is correct? ...
40
votes
6answers
3k views

Why don't analysts do category theory?

I'm a mathematics student in abstract algebra and algebraic geometry. Most of my books cover a great deal of category theory and it is an essential tool in understanding these two subjects. Recently, ...
18
votes
3answers
1k views

When do weak and original topology coincide?

Let $X$ be a topological vector space with topology $T$. When is the weak topology on $X$ the same as $T$? Of course we always have $T_{weak} \subset T$ by definition but when is $T \subset ...
16
votes
1answer
698 views

Medial Limit of Mokobodzki (case of Banach Limit)

A classical Banach limit is very usefull concept for me, but there is a problem with the integration and even with the measurability, this means for a sequence $(f_n)_{n\in \mathbb{N}}$ of measurable ...
14
votes
2answers
879 views

The space of Riemannian metrics on a given manifold.

For a finite-dimensional smooth (Hausdorff, second-countable) manifold $M$, consider the set $$\mathcal{Met}(M) = \{ g : g \text{ is a Riemannian metric on }M \}.$$ I'd like to know about the typical ...
10
votes
2answers
949 views

Gelfand-Naimark Theorem

The Gelfand–Naimark Theorem states that an arbitrary C*-algebra $ A $ is isometrically *-isomorphic to a C*-algebra of bounded operators on a Hilbert space. There is another version, which states that ...
37
votes
1answer
974 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 ...
9
votes
2answers
884 views

$C_c(X)$ dense in $L_1(X)$

I did the following homework, can you tell me if my answer is correct? Prove that for any $\sigma$-compact, locally compact Hausdorff space $\Omega$ equipped with a Radon measure $\mu$ the set of ...
8
votes
2answers
458 views

Reinventing The Wheel - Part 1: The Riemann Integral [closed]

Preface The core of any notion of integral is some sort of weighted sum: $$\sum b\mu(A)$$ Depending on wether the domain or range is decomposed these split into Riemann and Lebesgue type ones: ...
5
votes
4answers
3k views

Is the space $C[0,1]$ complete?

In order to prove $C[0,1]$ is complete, my functional analysis book says: "It is only necessary to show that every Cauchy sequence in $C[0,1]$ has a limit". It goes on by supposing $\{x_n\}$ is a ...
16
votes
4answers
955 views

Example of different topologies with same convergent sequences

It's well known that for metric spaces the following is true Let $ X $ be a space with two different metrics $ d_1,d_2$ such that the two topological spaces $ (X,d_1),(X,d_2) $ have the same ...
9
votes
1answer
409 views

Distance minimizers in $L^1$ and $L^{\infty}$

If $H$ is a Hilbert space, we have the Hilbert Projection Theorem, which tells us that given a nonempty, closed, convex subset $K \subset H$, and a point $x \in H$, there is a unique point $y \in K$ ...
4
votes
1answer
531 views

Eigenvalues of matrix with entries that are continuous functions

For each $t \in [0,b]$, let $M(t)$ be an $n \times n$ matrix with entries $m_{ij}(t).$ The matrix $M(t)$ is invertible and positive-definite, so the eigenvalues of $M(t)$ exist and are positive for ...
3
votes
2answers
2k views

Why $C_0^\infty$ is dense in $L^p$?

Why $C_0^\infty$ is dense in $L^p$? Would you give me a simple proof or the outline of the proof?
3
votes
1answer
575 views

Banach space in functional analysis

Prove that a closed subspace of a Banach space is also a Banach space. Show that the linear space of all polynomials in one variable is not a Banach space in any norm.
17
votes
1answer
521 views

Is $[0,1]^\omega$ homeomorphic to $D^\omega$?

Let $n\in \mathbb N$ and let $D^n$ be the closed $1$-ball in $\left(\mathbb R^n, \|\,\cdot\,\|_1\right)$. It is not too hard to show that $[0,1]^n \cong D^n$ in this case. This observation leads to ...