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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2
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1answer
105 views

Continuous mapping between $X$ and $C([0,1])$

Let X be a Banach space.Prove that a linear map $M:X \rightarrow C([0,1])$ is continuous iff for every $t\in [0,1]$, the rule $x \mapsto (Mx)(t)$ definies a continuous linear functional on X. My try: ...
4
votes
1answer
137 views

Questions about operator norm

I'm reading about functional analysis and I found the definition of the operator norm, if you have $(X,\|\|_1)$ and $(Y,\|\|_2)$ normed spaces then the set $\mathcal{L}_{\|\|_1,\|\|_2}(X,Y) := \{T:X ...
5
votes
2answers
282 views

Brownian Motion Covariance: max instead of min

It is known that $\operatorname{Cov}(B_t,B_s)=\min(t,s)$ where $B$ is Brownian motion. Can one think of an Ito process or integral (preferrably plain Gaussian process) $W$ such that ...
1
vote
1answer
111 views

Banach Spaces with Separable (Continuous) Duals are Separable. A clarification.

I'm aware of this post, but I have not yet been granted permission to comment on these other posts, so I hope you'll excuse asking for a clarification here. For clarity, I'll restate the question. If ...
5
votes
3answers
174 views

Find the necessary and sufficient conditions on $A$ such that $\|T(\vec{x})\|=|\det A|\cdot\|\vec{x}\|$ for all $\vec{x}$.

Consider the mapping $T:\mathbb{R}^n\mapsto\mathbb{R}^n$ defined by $T(\vec{x})=A\vec{x}$ where $A$ is a $n\times n$ matrix. Find the necessary and sufficient conditions on $A$ such that ...
9
votes
1answer
930 views

Why is $L^1(\mathbb{R}^n) \cap L^2(\mathbb{R}^n)$ dense in $ L^2(\mathbb{R}^n)$?

In Lieb and Loss's Analysis, I saw that they mentioned $L^1(\mathbb{R}^n) \cap L^2(\mathbb{R}^n)$ dense in $ L^2(\mathbb{R}^n)$ (dense wrt the $L^2$ norm, I think). But I didn't find its proof in the ...
5
votes
1answer
171 views

Bounded and compact sets in a subspace of $\mathbb R^{\mathbb N}$

Let $$ X= \{u=(u_1, u_2, \ldots): u_n \ne 0 \text{ only for a finite number of terms}\}\subseteq\mathbb R^\mathbb N, $$ with the topology inherited from $\mathbb R^\mathbb N$ (the "pointwise ...
3
votes
1answer
195 views

When is the composition operator assigned to a measure-preserving map unitary?

Let $(X,\mathcal{B},\mu)$ be a standard probability space, and let $T:X\rightarrow X$ be a measurable, measure-preserving transformation, i.e. for every $A\in\mathcal{B}$, $\mu(T^{-1}(A))=\mu(A)$. ...
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 ...
4
votes
0answers
157 views

What is the dual space of $C([0,T];X)$ ($X$ Hilbert space)?

What is the dual space of $C([0,T];X)$, where $X$ is a Hilbert space? Is it $\operatorname{BV}([0,T]; X^*)$? As we know, for $C([0,T])$, the dual space is $\operatorname{BV}([0,T])$, but when it is ...
6
votes
1answer
109 views

Uniform limit of finite-rank operators with the same rank.

Let $\{T_n\in\mathcal{B}(X)\,|\,\text{rank}(T_n)=R\,\}^{\infty}_{n=1}$ is a sequence of linear bounded finite-rank operators on a Banach space with the same rank $R$. Let it converge uniformly to an ...
3
votes
2answers
94 views

Compactness of operator $M: C([0,1]) \rightarrow C([0,1])$

Let $M: C([0,1]) \rightarrow C([0,1])$ be defined by $$ Mf(x) = f(x/2), \;\; x\in[0,1]$$ Is this operator compact? I have trouble using limit in operator norm of compact operator, or cauchy ...
3
votes
0answers
57 views

Prove that for every compact $K \subset \{z\in\mathbb{C}:|z|<1\}$ there is operator $T$ on Hilbert space such that $\sigma(T) = \sigma_p(T) = K$ [duplicate]

Possible Duplicate: Compact sets as point spectrum of a bounded operator Prove that for every compact set $K \subset \mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$ there is operator $T$ on Hilbert ...
3
votes
1answer
113 views

question about the definition of linear functions/operators (domains)

Suppose $\Omega_s \subset \mathbb{R}^n$ is a compact subset for each $s \in [0,T]$. I have a linear operator $$p_t^s:H^1(\Omega_t) \to H^1(\Omega_s)$$ which maps functions on $\Omega_t$ to functions ...
3
votes
1answer
82 views

Range of operator $ Mf(x) = f(x/2), \;\; x\in[0,1]$

Let $M: C([0,1]) \rightarrow C([0,1])$ be defined by $$ Mf(x) = f(x/2), \;\; x\in[0,1]$$ Prove that the range of $I-M$ does not contain nonzero constant functions, but it contains all functions ...
2
votes
0answers
74 views

Bound for a Sobolev function in an integral

For a compact bounded set $\Omega$, for the expression $$\int_\Omega \Delta u (\nabla u \cdot \nabla f)$$ where $u \in H^2$ and $f \in C^\infty$, is it possible to show that the expression is $\geq$ ...
4
votes
1answer
117 views

bounded operator between continuous functions

Let $M: C([0,1]) \rightarrow C([0,1])$ be defined by $$ Mf(x) = f(x/2), \;\; x\in[0,1]$$ Show that $M$ is bounded and that its spectrum is containd in the closed unit disc $\{ \lambda \in \mathbb{C} ...
6
votes
2answers
509 views

Do weak convergence and convergence of norms imply convergence in $L^2$?

Let $(f_n)_n\subseteq L^2(0,1)$ s.t. $$ f_n \rightharpoonup f, \qquad\qquad \Vert f_n\Vert_2 \to \Vert f\Vert_2 $$ where $\rightharpoonup$ means weak convergence. Is it true that $f_n \to f$ ...
5
votes
4answers
86 views

$\ell x = 0$ for all $\ell \in X'$ for banach spaces

I guess this is probably asked before but I can not find it. Let $X$ be a Banach space, and let $\ell x = 0$ for all $\ell \in X'$. Then $x = 0$ If all projections $\pi_\alpha x = 0$ and hence get ...
2
votes
1answer
482 views

Existence of solution for this parabolic PDE

The parabolic PDE $$\langle u', v \rangle + a(u,v) = \langle f, v \rangle \tag{*}$$ has a unique solution $u \in L^2(0,T; H^1)$ with $u' \in L^2(0,T;H^{-1})$ if $a$ is a bounded and coercive bilinear ...
1
vote
0answers
213 views

convolution of L1 function with a harmonic oscillation

I have to show that the convolution of a function $f \in L^1(\mathbf{R})$ with the harmonic oscillation $\phi_\omega (t) = \exp(2 \pi i t \omega)$ is equal to the Fourier Transform of $f$, ...
0
votes
1answer
63 views

Subspace in $I-T$ for bounded linear maps

Let $X$ be a Banach space and let $T:X \rightarrow X$ be a bounded linear map.Show that the range of $I - T$ contains the subspace $$Y_T = \{x \in X: \limsup_{n\rightarrow \infty} n^2\|T^nx\| < ...
3
votes
2answers
407 views

The transpose in Banach spaces is bounded below

Let $X$ be a Banach space and let $T:X \rightarrow X$ be a bounded linear map. Show that: If $T$ is surrjective then its transpose $T':X' \rightarrow X'$ is bounded below. My try: We know that ...
1
vote
1answer
148 views

Normal convergence

I have some problems to apply normal convergence of series of functions in any vector space. In fact $(f_{n})$ is a sequence of differentiable functions defined from a topological space $X$ to a ...
1
vote
1answer
133 views

Homework for Hahn-Banach theorem.Boundedness of $\overline {f}$

If X is a normed linear space ,$x_{1},x_{2},…,x_{n}\in X$ are linear independent,$a_{1},a_{2},…,a_{n}\in F$ are arbitrary,then there exist $f\in X^{\ast } $ such that $f\left( x_{k}\right)= ...
2
votes
1answer
277 views

Spectrum of a multiplication compact operator

Let $X=(C([0,1]), \Vert \cdot \Vert_{\infty})$. Determine the spectrum of $$ \begin{split} M \colon & X \to X\\ & u(t) \mapsto \int_0^t h(s)u(s)ds \end{split} $$ where $h \in C([0,1])$ ...
1
vote
2answers
483 views

Annihilator of a subset in the dual space.

Im reading Peter Lax book and he says: For any subset $S \subset X'$, we define $S^\perp$ as the subset of those vectors in $X$ that are annihilated by every vector in S. This confuses me a bit, ...
4
votes
1answer
293 views

Bounded linear maps in Banach spaces

Let $X$ be a Banach space and let $M: X \rightarrow X$ be a linear map. Prov that M is bounded iff there exists a set $S \subset X'$, dense in X', such that for each $\ell \in S$ the functional $m_l$ ...
1
vote
1answer
56 views

Existence of the right derivative of a function from the reals into a Banach space.

I am interested in knowing if the following is true: for any two vectors $v$ and $c$ in a Banach space over the reals : $ \lim_{h \to 0^{+}} \| \dfrac{v}{h} + c \| - \| \dfrac{v}{h}\|$ exists; the ...
2
votes
1answer
96 views

Closed extensions in the weak* topology

Let $\ell^\infty$ be the Banach space of bounded sequences with the usual norm. and let $\ell_0(x) = \lim_{n \rightarrow \infty} x_n$, for convergent sequences. Show that the sett L consisting of all ...
2
votes
2answers
219 views

the dimension of continuous functions on a compact set is finite [duplicate]

Possible Duplicate: vector space of continuous functions on compact Hausdorff space This is a problem am trying to solve. Suppose the dimension of $C(X)$ is finite where $X$ is compact ...
0
votes
1answer
117 views

How to find subdifferential of $|x|$

I want to compute the subdifferential of $ f $ on $ \mathbb{R} \setminus \{ 0 \} $ when $ f(x) = |x| $. How do I do this?
1
vote
0answers
157 views

$f$ is concave and convex in its arguments.

Suppose that X be a reflexive Banach space and function $f:X×X↦R$ which is concave in its first argument and convex in its second one. How to prove $f(x,x)=0$ for all $x∈X$?
3
votes
1answer
58 views

How to prove that $\lim_{n\to\infty} n^{-1} \ln (\|A^n\|)=\ln(\rho(A))$, where $A$ is a matrix.

How to prove the following fact or where can I find its proof? Let $A \in M(\mathbb{R}^d)$ and assume that $\rho(A)>0$ (where $\rho(A)$ stands for the spectral radius of $A$). Then, for each ...
7
votes
3answers
178 views

why do successive eigenfunctions have more oscillation?

I was told the following argument as to why successive eigenfunctions tend to have more oscillations: Suppose (without worrying about why) that the first eigenfunction has the least oscillation. ...
5
votes
1answer
497 views

Norm in a dual space

If $f \in X^*$, with $X^*$ the dual space consisting of all linear bounded functionals on a linear normed space $X$. With the norm defined as $||f||_{X^{*}} = \sup_{||x|| \leqslant 1} |f(x)|$. Why ...
1
vote
1answer
1k views

Supremum of continuous functions is continuous?

If $f(t,u)$ is continuous wrt. $t$ (and $u$), then is $$\sup_{u \in H^1(\Omega)} f(t,u)$$ continuous wrt. $t$? I am unable to prove this. Help appreciated.
2
votes
3answers
130 views

Fixed point in a continuous map [duplicate]

Possible Duplicate: Periodic orbits Suppose that $f$ is a continuous map from $\mathbb R$ to $\mathbb R$, which satisfies $f(f(x)) = x$ for each $x \in \mathbb{R}$. Does $f$ necessarily ...
3
votes
1answer
104 views

Equicontinuity and uniform boundedness for “distributions”

Exercise. (Rudin, Functional Analysis, chapter 2, pag. 53). Let us consider the space $$ \mathcal D :=\{f \in C^{\infty}(\mathbb R), \, \text{supp}f\subseteq [-1,1] \} $$ with the topology induced by ...
2
votes
1answer
166 views

An inequality for Holder norms

Let $f$ be a non-negative $C^2$ function on a compact domain $\Omega$ in $\mathbb R^n$. I am trying to prove the inequality $$\|\sqrt f\|_{C^{0,1}(\Omega)}\leq C(1 + \|f\|_{C^{1,1}(\Omega)})$$ where ...
3
votes
2answers
636 views

Show that a finite-dimensional Banach space has a bijective compact operator

It is clear that if $ T: X \rightarrow X $ is a bijective compact operator, where $ X $ is a Banach space, then $ \dim(\text{Range}(T)) = \dim(X) $, which implies that $ \dim(X) $ must be $ < ...
6
votes
1answer
587 views

Uniqueness for 3-dimensional heat equation initial Robin boundary value problem (SOLVED)

Let $\Omega \subset \mathbb{R}^3$ be a bounded domain. Using an energy argument, show that the IBVP \begin{align} u_t &= \Delta u ~~~~~~~~~~x \in \Omega, ~t>0\\ \frac{\partial u}{\partial \nu} ...
7
votes
1answer
366 views

Weak convergence in $L^p$ and uniform convergence

I don't understand the last line of a proof (which is supposed to be obvious...), could you help me? The context is the following. We have a bounded open set $U$ of $\mathbb{R}^m$, and a ...
2
votes
1answer
302 views

Proving that an operator is bounded from below

Hi again Stackexchange, This is a question I had on an exam today that I could not answer, and it deeply disturbs me. Let $X$ be a Banach space and $T:X\to X$ a surjective linear map. Show that the ...
3
votes
0answers
220 views

Borel - Caratheodory Inequality

If $f$ is a complex-valued function analytic on $\{z:\vert z \vert \leqq r \}$, then for $\vert z \vert <r $, $$ \vert f(z) \vert \leqq \frac{2\vert z \vert}{2-\vert z \vert} \sup\{\Re f(w): \vert ...
5
votes
1answer
210 views

Compact linear operator from $L^p (\mathbb R)$ to $L^p (\mathbb R)$

Wanted to prove the following question since one week but couldn't get even single idea on it . Here is the question : if $m : \mathbb R \to \mathbb R$ a measurable function $1 \le p <\infty $ and ...
5
votes
1answer
93 views

Pseudonormable Product Spaces

I want to prove that a product $\prod_{i\in I}X_i$ of topological vector spaces is pseudonormable only if a finite number of the factor spaces are also pseudonormable and the rest have the trivial ...
1
vote
1answer
91 views

A question about the second differential

Hi I have a doubt: What is the matrix associated at the second differential? Let $f:\mathbb{R}^n \rightarrow \mathbb{R}$, differiantable and let $df: \mathbb{R}^n \rightarrow \mathbb{R}$, ...
2
votes
0answers
96 views

Convergence of Schwartz kernels implies convergence of operators

Let $K$ be a smoothing operator on $\mathbb{R}^n$, i.e., it defines a map on all Sobolev spaces $K\colon H^r(\mathbb{R}^n) \to H^s(\mathbb{R}^n)$ for all $r, s \in \mathbb{R}$. Now (a variation of) ...
0
votes
1answer
34 views

Uniformizability of a space

Let $E$ be a topological space. For $x \in E$,the nhds of $x$ which are both closed and open form a fundamental system of nhds of $x$.Show that E is uniformizable. Check here for definition of ...