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, ...

learn more… | top users | synonyms (1)

2
votes
1answer
132 views

Understanding the definition of the generator of a semigroup of operators

Given a strongly continuous semigroup $T : \mathbb{R}_+ \to L(B)$ on a Banach space $B$, its infinitesimal generator $A$ of a strongly continuous semigroup $T$ is defined as a mapping $B \to B$ as $$ ...
1
vote
1answer
58 views

Prove that $ A\subset \ell_1 $ is compact iff $A$ satisfies the following property

$A$ is compact iff $ A $ is bounded and, given $\epsilon > 0$, there exists $ n_0$ such that $ \sum_ {k=n}^\infty |x_k|\le\epsilon $ for all $n \geq n_0 $ and for all $ x\in A $. To prove ...
2
votes
1answer
187 views

Question regarding Gâteaux Derivative

This is a question on an assignment for a grad engineering class that I cannot seem to figure out. The statement is as follows: Consider $X$ the space of continuous functions on the interval [0,1]. ...
1
vote
1answer
482 views

Inverse fourier transform 3 dimensions

Hoi, I want to show that for $n=3$ that $$\mathcal{F}^{-1}\left(\frac{1}{1+|s|^2}\right) = \frac{1}{4\pi |x|}e^{-|x|} $$ As a hint I've been given: Its the unique solution to the equation ...
3
votes
2answers
264 views

Extracting a subsequence from a sequence of $\mathcal{L}^1$ functions

Any help with the following problem is appreciated. Given: a sequence of nonnegative functions $(g_n)$ which are U.I. (uniformly integrable) in $\mathcal{L}^1(0,1)$ with $\sup_n \Vert g_n \Vert_1 ...
3
votes
1answer
256 views

Invertibility of compact operators

I'm a little confused about compact operators and whether or not they are invertible. Just hoping someone here can clear up my confusion: Let $T$ be a compact operator on a Banach space $X$. Since ...
5
votes
1answer
558 views

Using Lax Milgram to find a weak solution in an intersection of Sobolev spaces

I am trying to prove the existence of a weak solution of the problem: $$ -\Delta^2 u = f \in L^2(U)\\ \\ u|_{\partial U}=\Delta u|_{\partial U} = 0 $$ on the bounded open set $U\subset\mathbb{R}^n$ ...
2
votes
3answers
80 views

Clarification of sequence space definition

Let $(x_n)$ denote a sequence whose $n$th term is $x_n$, and $\{x_n\,:\,n\in\mathbb{N}\}$ denote the set of all elements of the sequence. I have a text that states Note that ...
4
votes
1answer
161 views

Existence of a non zero element in the dual

Let $S$ a vector subspace of a normed vector space $X$ such that $\overline{S} \neq X$. Show that, with the Hahn-Banach Theorem (Geometric Version), that there is $F\in X^{\prime}$ such that ...
1
vote
1answer
752 views

Dual Space as a Hilbert Space

I have this problem: Let $(X, \langle\cdot,\cdot\rangle)$ a Hilbert Space on $\mathbb{R}$ with Riez map $\mathcal{R}:X^{\prime}\rightarrow X$, define $[\cdot,\cdot]:X^{\prime}\times ...
2
votes
2answers
143 views

Hahn-Banach Separation Theorem and Bishop's Theorem

I am looking at the proof of Bishop's Theorem on pages 122 and 123 of Rudin's Functional Analysis. The following quote is from the the last two sentences of the proof on pg. 123. "Every continuous ...
2
votes
1answer
78 views

Simple doubt about dual norm

If $(X, \|\cdot\|)$ is a normed vector space, then $$\|F\|_{X^{\prime}}\ =\ \sup_{x\in X-\{0\}}\frac{|F(x)|}{\|x\|},$$ by definition. Then I want prove that, $$\|F\|_{X^{\prime}}\ =\ ...
3
votes
1answer
248 views

Stone-Čech compactification. A completely regular topological space is locally compact iff it is open in its Stone-Čech compactification.

I would like to show that a completely regular topological space is locally compact iff it is (weak-star) open in its Stone-Čech compactification. Does this hold in general? I.e given a compact ...
3
votes
1answer
110 views

$K$ is a linear compact operator on Hilbert space $H$. Will the image of $I-K$ on every closed subspace of $H$ be also closed?

Just as the title. We know the image of $I-K$ is closed, but if we restrict $H$ to a closed subspace $V$, will $(I-K)(V)$ be a closed subspace of $H$? Any hint is appreciated.
1
vote
1answer
142 views

Multiplication of distributions by smooth functions

Let $u\in D'(\mathbb{R})$ and $f\in C^{\infty}$. I'm trying to figure which of the following statements is true: I. If $f\restriction_{supp(u)}=1$ then $f\cdot u=u$. II. If ...
3
votes
2answers
270 views

Proving Bishop's Theorem using Krein-Milman Theorem

I am studying the proof of Bishop's theorem (generalization of Stone-Weierstrass) in Rudin's Functional Analysis 2nd edition. He make the following statement on the bottom of page 122, "Since $\mu ...
0
votes
2answers
657 views

orthogonal complement of symmetric matrices

How do I can prove that the orthogonal complement of space of symmetric matrices is the space of skew-symmetric matrices? With the inner product $\langle A,B\rangle = \mbox{tr}(A^TB)$. Thanks in ...
5
votes
2answers
285 views

Is the weak-star topology on the dual of a Banach space completely regular?

Does the weak-star topology on the dual of a separable Banach space make the dual completely regular under weak-star topology? So I have come to the stage in a proof where if I could show this, then ...
0
votes
1answer
240 views

Is the unit ball of a separable Banach space itself separable?

If $X$ is a separable Banach space, then do we know that its unit ball has a countably dense subset contained in the unit ball? This isn't obvious to me.
1
vote
1answer
53 views

Question on notation: What does $0 \leq M \leq 1$ mean for a bounded operator $M$?

Let $\mathcal{H}$ be a Hilbert space and let $M\colon \mathcal{H} \rightarrow \mathcal{H}$ be bounded linear operator. I am working through a paper by Roger Godement from the 1950's. In one section ...
5
votes
4answers
2k views

Orthogonal complement of a Hilbert Space

I have this problem: Let $S$ be a subset of a Hilbert $H$ and let $M$ be the closed subspace generated by $S$. Show that $M^{\perp} = S^{\perp}$ $M = (S^{\perp})^{\perp}$ if $V$ is a subspace of ...
4
votes
1answer
935 views

Unit ball of a Separable Banach Spaces is metrizable

Please help me to understand why the unit ball of a separable banach space is metrizable, when it is given the induced topology from the weak topology. Specifically, my problem is I think I'd be able ...
1
vote
0answers
107 views

Maximal Ideals of the Wiener Algebra

I'm wondering why the maximal ideals of the Wiener algebra $\mathcal{W}$ are of the form $\{M_z:z\in \mathbb{T}\}$ where $M_z=\{f\in \mathcal{W}\; |\; f(z)=0\}$. Given that the Wiener algebra is a ...
0
votes
1answer
77 views

convergence of product

Let $I$ an interval in $\mathbb{R}$ Let $f_n$ bounded in $H^1(I),$ then we can extract a subsequence such as $f_n \rightarrow f$ strongly in $L^2(I)$ 2- Let $g_n$ bounded in $L^{\infty}(I)$ , ...
1
vote
1answer
113 views

Suppose that $(X,\|\cdot\|)$ is a separable Banach space. Then the unit ball of $X'$ is a $Z$-set in the weak-star topology.

Suppose that $(X,\|\cdot\|)$ is a separable Banach space. I want to show that the unit ball $B'$ of $X'$ is a $Z$-set in the weak-star topology. Meaning that there exists $f \in C_b(X')$ such that $B' ...
3
votes
1answer
83 views

Lipschitz Continuity of Optimal Value

Let $m$ be a probability measure on $W \subseteq \mathbb{R}^p$, so that $m(W) = 1$. Consider a function $f : X \times Y \times W \rightarrow [0,1]$, where $X \subset \mathbb{R}^n$, and $Y \subset ...
3
votes
0answers
75 views

(DFM) vs (DFS) spaces, Banach scales

I have already posted this question on MO: http://mathoverflow.net/questions/126007/dfm-vs-dfs-spaces-banach-scales However, not having received any feedback, I decided to repost it, since I have the ...
1
vote
0answers
83 views

Subspace of $L^2(U)$ that are closed under multiplication?

Suppose $U \subset \mathbb{R}^n$ is non-empty, bounded, open, connected with $C^1$ boundary. Is there any subspace of $L^2(U)$ that are closed under multiplication, that is under which conditions can ...
2
votes
1answer
151 views

(p-q)-Lipschitz continuity of linear function

I have the following linear function $f(x,y,z) = ax + by + cz.$ I need to prove that f() is (p-q) Lipschitz continuous where $p=1$ and $q=\infty$. For a given two points $(x_1, y_1, z_1)$ and $(x_0, ...
3
votes
1answer
340 views

Positive functions with zero integrals

I was a bit confused by this link mentioned in this question - in particular, in Remark 4.21: Suppose that $f$ is a positive function on $[a,b]$. If $f$ is Henstock-Kurzweil integrable, then the ...
2
votes
1answer
126 views

Is it true that for every subspace $N$, we have $N^{{\perp}{\perp}}=N $?

Let $ ‎X=C[-1,1]‎$‎‎ be inner product space with definition $$‎\langle f,g‎‎‎\rangle =‎\int_{-1}^1 f‎‎ \overline{g}‎ ‎dt ‎‎.$$ Let $M$ be the subspace defined by ‎$$ ‎M= ‎‎\left\{f‎ \in ‎X\mid ...
2
votes
1answer
114 views

Proof of implication: $\varphi^*\text{ is bounded below}\implies\varphi\text{ is a quotient map}$

We say that a bounded operator $\varphi:X\to Y$ is $c$-topologically injective if $\Vert\varphi(x)\Vert\geq c\Vert x\Vert$ for all $x\in X$ $c$-topologically surjective if for all $y\in Y$ there ...
1
vote
1answer
64 views

Example for a sequence of functions in $\mathcal{L}^1[0,1]$

I am looking for an example of a sequence of functions $(g_n)$ that is in $\mathcal{L}^1[0,1]$ and U.I. so that the following three conditions are satisfied: $\forall \, n\, \, \vert g_n \vert ...
1
vote
1answer
129 views

Calculation of the Laplacian of a function in $\mathbb{R}^3$.

I have to calculate the Laplacian distributional sense) of the following function $$\frac{e^{i\sqrt{\lambda}|x|}-1}{4\pi|x|}$$ with $\lambda>0$, $x\in\mathbb{R}^3$. I've procedded in the following ...
4
votes
0answers
58 views

A normed space is not separable if and only if it contains an uncountable set of disjoint balls of the same radius

I want to show normed space $E$ is not separable iff $E$ contains uncountable set of pairwise non-intersecting balls of radius $r > 0$. Use contraposive: first prove $E$ is separable then ...
2
votes
1answer
51 views

Regularity and the Varitational Inequality

Let $K = \left\{ v \in H_0^1(\Omega) \, : \, v \geq 0 \right\}$, further suppose $\Omega$ has the regularity property that $||v||_{H^2} \leq C(\Omega)||\Delta v||_{L^2}$, for all $v \in ...
0
votes
2answers
114 views

Boundedness of a Solution Operator

Let $v \in L^2(\partial \Omega)$ and define $S(v) := y$ where $y$ satisfies $-\Delta y + y = 0 $ in $\Omega$ and $\frac{\partial y}{\partial \nu} = v$ on $\partial \Omega$. I need to show that $S$ is ...
-1
votes
2answers
206 views

Weak star limit

Let $\Omega = \mathopen]0,1\mathclose[$ and let a function $A_n: \Omega \to \mathbb R$ defined as: $$A_n(x) = \begin{cases}\alpha &\text{if } k \epsilon \leq x < (k+\tfrac{1}{2}) \epsilon \\ ...
2
votes
1answer
317 views

Are the pre-compact sets in a locally convex space with the weak topology exactly the bounded sets?

The Wikipedia article on totally bounded space claims that "In a locally convex space endowed with the weak topology the precompact sets are exactly the bounded sets." The claim is repeated on ...
3
votes
2answers
276 views

$l_1$ equipped with the sup norm is NOT a Banach Space

Prove that $l_1 = \{ x = (x_k)_{k\in\mathbb{N}}\subset \mathbb{R};\ \sum_{k\in\mathbb{N}}\ |x_k| < +\infty \}$ equipped with the norm $\| x\| = \mathrm{sup}_{k\in\mathbb{N}} |x_k|$ is NOT a Banach ...
2
votes
0answers
112 views

Baire $\sigma$-algebra generated by Z-sets.

Given a completely regular Hausdorff topological space $(X,\tau)$ I want to be able to show that the Baire $\sigma$-algebra generated by the Z-sets is the same as the smallest $\sigma$-algebra for ...
1
vote
1answer
277 views

Basic question on bounded linear operators in Banach spaces.

I have sincerely tried this problem, for way too long, and I must admit defeat. How am I to prove the following? Let X be a Banach space and I be the identity mapping on X. If T is a bounded linear ...
3
votes
1answer
83 views

About the property of Littlewood-Paley partition of unity.

Let $\{\phi_j\}_{j=0}^\infty$ be the Littlewood-Paley partition of unity, i.e., $$ \sum_{j=0}^\infty \phi_j(\tau) = 1 \; for \; \tau \geq 0; \;\;\;\;\phi_j \in C_0^\infty (\Bbb R), \;\phi_j \geq 0 \; ...
-1
votes
1answer
109 views

Showing $h(x) = \frac{1}{\epsilon}\int_x^{x+\epsilon} f(t)dt$ is differentiable for continuous $f$ and $\epsilon > 0$

Assume $f$ is continuous on $\mathbb{R}$ and $\epsilon>0$. Let $h(x) = \displaystyle\frac{1}{\epsilon}\int_x^{x+\epsilon} f(t)dt$. Show $h$ is differentiable and $h'$ is continuous. Compute ...
2
votes
1answer
65 views

Extending a distribution continuously to $C_c^N (\Omega)$

Let $\Omega \subseteq \mathbb{R}^n$ be a domain and let $u\in D'(\Omega)$ be a distribution of order $\leq N$. How can we show that $u$ can be continuously extended to $C_c^N(\Omega)$? By ...
2
votes
1answer
100 views

metric projection onto one dimensional (closed ) subspace in $L_p (p\neq 2)$

I want to know "if the metric projection onto one dimensional (closed) subspace in $L_p (p\neq 2)$ is linear? I think it is not linear, but I can not give a strict proof. Thanks for any answer!
0
votes
1answer
621 views

Show that for any partition P of $[a,b]$, $U(f,P) - L(f,P)$ $\leq$ C(b-a)mesh(P)

Suppose f:[a,b]-> R is Lipschitz, i.e |f(x)-f(y)| <= C|x-y| for all x,y in [a,b] and thus f is continuous, Show that for any partition P of [a,b], U(f,P) - L(f,P) $\leq$ C(b-a)mesh(P). Some ...
6
votes
1answer
240 views

Exercise Functional Analysis

Let $\mathcal{F}$ be the set of all functions $f: \mathbb{R} \rightarrow \mathbb{R}$. Consider an operator $\mathcal{O}: \mathcal{F} \rightarrow \mathcal{F}$ such that: $\mathcal{O}( f_1 + f_2) = ...
4
votes
0answers
231 views

Inverse of Identity plus Volterra operator

consider the following operator or $L_2(0,1)$, $(Pw)(x)=w(x)+\int_0^x K(x,y)w(y)dy+\int_x^1 K(y,x)w(y)dy$, where the integral kernel is a polynomial. I am trying to construct the inverse of this ...
4
votes
1answer
240 views

Hilbert space and its dual

I have an elliptic equation in the form $$-\Delta u + u =F(u).$$ For any $\phi \in C^{\infty}_{0}$ we rewrite the elliptic equation in weak form $$\int \limits_{\mathbb{R}^n}\left(-\Delta u + ...