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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6
votes
2answers
513 views

equivalence of norms

I would like a little help here: I have two defined norms over $C^{1}([0,1])$ : $\| A(f)\|=|f(0)|+\max_{x\in[0,1]}{|f'(x)|}$ $\| B(f)\|=\int_0^1|f(x)|dx+\max_{x\in[0,1]}{|f'(x)|}$ I already ...
0
votes
1answer
971 views

Lebesgue integrability of continuous function in closed interval

I'm trying to show that a continuous function $f$ in $[a,b]$ is Lebesgue integrable, using approximation through step functions. It is pretty trivial to show using the connection between Riemann ...
1
vote
2answers
87 views

Find a optimal Sobolev Space $H^s$

It is a homework problem. Function $f:\mathbb{R}^2\rightarrow\mathbb{R}$,$\Omega$ is a disk at origin with radius as $\dfrac{1}{2}$. $f(x) = \log(\log(\dfrac{1}{|x|}))$, where $x\in\Omega$(i,e. ...
2
votes
1answer
114 views

Subspace of $\ell_\infty$ that is not separable

I need to prove that $$L = \left\{ (x_i)\in \ell_\infty : \lim_{n\to\infty} \frac{1}{n} \sum_{i=1}^n x_i = 0\right\}$$ is a subspace in $l_\infty$ (done that), is closed in $l_\infty$ (done ...
4
votes
1answer
248 views

Isometric Isomorphism from $c_0$ to $C[0,1]$

I need to prove that there is an isomorphic isometry from $c_0$ to some subspace of $C[0,1]$. Researching a bit, it looks like it follows from Banach-Mazur theorem, but we haven't studied it, at least ...
1
vote
1answer
90 views

Convergence in norm operator

If I have an operator valued functions $A(z):H_1\to H_2$ such that the following limit $$\lim_{z\to z'}A(z)=A(z')$$ exists in the uniform topology of $B(H_1,H_2)$, that is $$\Vert ...
1
vote
1answer
50 views

Inequalities for point distribution

Let $u$ be a distribution on $\mathbb{R}^n$ with support = $\left\{0\right\}$. Then there exists $N$ such that $u$ has order $N$. Let $\chi\in C_0^{\infty}(\mathbb{R}^n)$ a smooth function with ...
2
votes
1answer
81 views

Equilateral triangle in $l_p^2$

I was trying to calculate the side of an equilateral triangle with the vertices on the unit sphere of $l_p^2$, when $1<p<\infty$. When I say "equilateral", I mean with respect to the distance in ...
1
vote
1answer
127 views

Relationship between eigenvalues of a matrix and its derivative

Is there any relationship between the derivative of a matrix which depend by a parameter and its eigenvalues?
1
vote
1answer
217 views

A question on locally integrable function

Let $f$ be a locally integrable function, $f\in L_{\operatorname{loc}}^{1}(\mathbb{R}^n)$. Prove that the operator $$T_f:\phi\to\int_{\mathbb{R}^n}f(x)\phi(x)dx$$ is a distribution. (See ...
3
votes
1answer
77 views

Are these norms equivalent?

I have to prove thate these norms are equivalent $$\Vert f\Vert_1^2=\int_{\mathbb{R}^3}d^3k\sqrt{1+\vert k\vert^2}\vert\hat{f}(k)\vert^2$$ and $$\Vert ...
4
votes
1answer
99 views

Dual Space of space of weighted functions

Let $g: \mathbb{R} \to \mathbb{R}$ be a positive function uniformly bounded away from $0$. Let $C(\mathbb{R})$ be the space of continuous functions that with norm $| f | := \sup_{x \in \mathbb{R} } ...
1
vote
1answer
262 views

Holder Convergence of the Composition of Holder and Lipschitz Continuous functions

Trying to fill in a proof, and I was wondering if the following is true. Let $f \in C^{\gamma}(\mathbb{[0,1]})$ (ie a $\gamma$-Holder continuous funtion with $\gamma \in (0,1)$). Define the norm $\| ...
3
votes
2answers
67 views

Proving that certain subspace of $\ell_1$ is non closed

I need to prove that $$L= \left\{(x_i) \in\ell_1 : \sum_{i=1}^\infty ix_i= 0\right\}$$ is non-closed in $\ell_1$. I can't really think of sequences of sequences that are in this subspace, much less ...
1
vote
1answer
131 views

Totally bounded set, $\varepsilon$-nets

I'm working through Martin Schechter's "Principles of Functional Analysis" (2nd ed.) and a problem concerning totally bounded subsets and $\varepsilon$-nets of normed linear vector spaces on page 96 ...
0
votes
3answers
181 views

Trying to prove that operator is compact

Consider $T\colon\ell^2\to\ell^2$ an operator such that $$T((x_n))=(2^{-n}x_n); \forall x=(x_n)\in \ell^2 $$ Does anyone know how to prove that it is compact? I understand that I have to find a ...
0
votes
1answer
84 views

Construction of an explicit isomorphism

Is there an explicit isomorphism between the Hilbert spaces: $L^2[0,1]$ and $\ell^2$, if the answer is 'yes' how do I construct it ? It is true to say that the Hilbert isomorphism must be in the same ...
1
vote
1answer
219 views

Closure of span in a Hilbert space.

I've got a functional analysis problem that I am not sure where to start with at the moment. Any help would be great. If A is an arbitrary subset of a Hilbert space $(X,\langle .,. \rangle)$, then ...
1
vote
1answer
68 views

Reference Request: Vector Spaces

I am a new student in the field of functional analysis. I'm looking for references that make sense for all kinds of vector spaces, such as the difference between $L^2$ and $l^2$ and others like: ...
8
votes
1answer
282 views

Short and elegant introduction to Sobolev spaces

I am preparing a course on Nonlinear Analysis, and I need to teach the most important facts about Sobolev spaces to my students. I know most books on this subject, from Brezis' to Adams', from Mazya's ...
2
votes
0answers
66 views

Invertibility of a matrix

I have to study the invertibility of this matrix $$A(k)=\bigg[\bigg(\alpha_j-\frac{ik}{4\pi}\bigg)\delta_{jj'}-\tilde{f}(y_j-y_{j'})\bigg]_{jj'}$$ where $\tilde{f}(x)$ is ...
5
votes
1answer
97 views

Why locally compact in the Gelfand representation?

I'm missing something in the Gelfand representation. Let's just say $\mathfrak{A}$ is a Banach algebra. Then it's a Banach space, and so we have $\mathfrak{A}^\ast$. The multiplicative linear ...
3
votes
2answers
1k views

Spectral radius of the Volterra operator

The Volterra operator acting on $L^2[0,1]$ is defined by $$A(f)(x)=\int_0^x f(t) dt$$ How can I calculate the spectral radius of $A$ using the spectral radius formula for bounded linear operators: ...
0
votes
1answer
39 views

Computing a derivative of map from $V \to V^*$ (PDEs and regularity)

I am reading Rogers and Renardy book on parabolic regularity. There they consider a PDE $$\dot u = A(t)u + f(t)$$ where $A(t):V \to V^*$ is an operator. In the regularity result, they need $A \in ...
2
votes
1answer
100 views

Calculation of the Fourier transform of a function

I have calculated the Fourier transform of this function $$f(x)=\frac{e^{i\sqrt{z}|x-y|}}{4\pi|x-y|}$$ with $x\in\mathbb{R}^3$, $\Im \sqrt{z}>0$ e $y$ fixed point in $\mathbb{R}^3$. I have obtained ...
0
votes
1answer
118 views

Prove that there is a subsequence of functions which converges uniformly

The problem is this Let $\phi:[0,1]\times\mathbb{R}\rightarrow \mathbb{R}$ be bounded and continuous, and for $n=1,2,\dots$ let $f_n:[0,1]\rightarrow \mathbb{R}$ satisfy $f_n(0)=1/n$ and ...
1
vote
0answers
52 views

Sum of Hilbert spaces

For various reasons I need to define the following space $$\hat{H}=\{u\mid u=f+g, \;f\in H^{2,-s}(\mathbb{R}^3),\; g\in L^{2,-s}(\mathbb{R}^3)\}$$ where ...
1
vote
1answer
32 views

Question about a proof.

I was just reading over this proof here Adjoint identity . I am just learn functional analysis, and I wonder, in the step where it says "Since $D(A)$ is a subspace we must have $f(v,Av)=0$", why this ...
3
votes
3answers
208 views

distribution with point support

Let $u$ be a distribution on $\mathbb{R}^n$ with support = $\left\{0\right\}$. Then there exists $N$ such that $u$ has order $N$. Let $\chi\in C_0^{\infty}(\mathbb{R}^n)$ a smooth function with ...
9
votes
2answers
2k views

How to show that quotient space $X/Y$ is complete when $X$ is Banach space, and $Y$ is a closed subspace of $X$?

How to show that quotient space $X/Y$ is complete when $X$ is Banach space, and $Y$ is a closed subspace of $X$? Here's my attempt: Given a Cauchy sequence $\{q_n\}_{n \in \mathbb{N}}$ in $X/Y$, each ...
10
votes
1answer
531 views

An approximate eigenvalue for $ T \in B(X) $.

This is a problem from Conway’s Functional Analysis: Definition An approximate eigenvalue for $ T \in B(X) $ is a scalar $ \lambda $ such that there is a sequence of unit vectors $ x_{n} \in X $ ...
6
votes
1answer
691 views

Isomorphic Hilbert spaces

As part of a broader proof , I need to show that every two separable Hilbert spaces (that contains a dense countable set) are isomorphic (the linear mapping from one space to the other is injective ...
1
vote
1answer
63 views

Find a decoupled explicit formula for a minimizer

Consider the energy $F(u,v) = \int^1_0((\frac{1}{4}(u')^2+(v')^2 +\frac{1}{2}(u-v+1)^2)dx$ for $C^1$ functions u and v on the interval (0,1) that satisfy the boundary conditions ...
3
votes
1answer
139 views

Is the unit ball: $B(0,1)=\{f \in L_p(X,u): \|f\|_p<1\}$ convex? , $0<p<1$

Let $(X,\mathbb{X},u)$ be a measure space $L_p(X,u)=\{ f:X\to \mathbb{C}: \|f\|_p<\infty\}$ , $0< p <1$ , $f$: measurable function $\|f\|_p=\left( \displaystyle \int_X |f|^p ...
4
votes
1answer
223 views

Normal $T\in B(H)$ has a nontrivial invariant subspace

I am wondering if the following is true: Every normal $T\in B(H)$ has a nontrivial invariant subspace if $\dim(H)>1$?
3
votes
1answer
190 views

The span of the orthorgonal projections is norm dense in $B(H)$

This is a question in my functional analysis book. How to use the spectral theorem to prove that the span of the orthogonal projections is norm dense in $B(H)$?
5
votes
0answers
65 views

$e^{iBt}e^{-iAt}$converges as operator norm

Let $A,B$ be self-adjoint operators on $H$,then we can define the strong limit $$ W=s-\lim_{t\to+\infty}e^{iBt}e^{-iAt} $$ If the limit exsists, then W is called the wave operator, which is ...
3
votes
1answer
158 views

Operator norm and spectrum

Let $L$ be a bounded linear operator on a Hilbert space. Without assuming finite dimensions, can we express the operator norm of $L$ in terms of the spectrum of the positive operator $L^{\dagger}L$? ...
2
votes
1answer
147 views

linear transformation between spaces of continuous function on metric space

Let $M_1$ and $M_2$ be compact metric spaces. Denote $C(M_i)$ as the space of continuous functions from $M_i$ to $\Bbb C$ with supremum-norm, $i=1,2$. A linear function $T:C(M_1)\to C(M_2)$ is said ...
6
votes
1answer
97 views

$L_{p}(\mathbb{T})$ is not uniformly convex if $p \in \{1,\infty\}$.

How can I prove that $L_{p}(\mathbb{T})$ is not uniformly convex if $p \in \{1,\infty\}$. Here $\mathbb{T} = \mathbb{R}/\mathbb{Z}$
3
votes
1answer
359 views

Polar decomposition of invertible elements in a unital C$ ^{*} $-algebra.

If $ A $ is a unital C$ ^{*} $-algebra and $ a $ is invertible, then $ a = u|a| $ for a unique unitary element $ u $ of $ A $. If $ \| a \| = \| a^{-1} \| = 1 $, what can you say about $ |a| $? I ...
11
votes
1answer
537 views

Motivation of Feynman-Kac formula and its relation to Kolmogorov backward/forward equations?

Kolmogorov backward/forward equations are pdes, derived for the semigroups constructed from the Markov transition kernels. Feynman-Kac formula is also a pde corresponding to a stochastic process ...
4
votes
3answers
405 views

Let $ (x_n) $ be a divergent sequence in a compact subset of $ \mathbb{R}^n $. Prove there are two subsequences that converge to different limits.

Let $(x_n)$ be a divergent sequence in a compact subset of $\mathbb R^n$. Prove that there are two subsequences of $(x_n)$ that are convergent to different limit points. Some ideas that might be ...
1
vote
1answer
168 views

Question on Continuous function and Lipschitz

$f:\mathbb{R}^n \to \mathbb{R}$ is contiuous. If $x \in \mathbb{R}^n$ and $C \in \mathbb{R}$ such that $f(x) < C$ ($C$ is constant). Prove that there is $r>0$ such that $\forall{y} \in B_r ...
2
votes
1answer
123 views

using uniform boundedness principle

I have a sequence of numbers $x_n$ that satisfy that for every $y_n \in c_0$ (when $c_0$ is a Banach space of all the complex sequences that satisfy $\lim_{n\rightarrow \infty }{a_n} =0$ ) the series ...
1
vote
1answer
84 views

Range of the generator of a one parameter semigroup of operators?

From Wikipedia: If $X$ is a Banach space, a one-parameter semigroup of operators on $X$ is a family of operators indexed on the non-negative real numbers $\{T(t)\} t ∈ [0, ∞)$ such that $$ ...
2
votes
1answer
626 views

Minimizing continuous, convex and coercive functions in non-reflexive Banach spaces

Let $X$ be a infinite dimensional real Banach space. If $X$ is reflexive, then any continuous, convex coervive function $f:X\rightarrow\mathbb{R}$ has a minimum value, that is assumed for some point ...
3
votes
1answer
229 views

Why is a strongly continuous one-parameter semigroup called a $C_0$-semigroup?

Why is a strongly continuous one-parameter semigroup called a $C_0$-semigroup? Isn't $C_0$ the set of continuous functions that vanish at infinity? Thanks and regards!
4
votes
2answers
185 views

Finding an explicit expression for a minimizer

Suppose $f$ is a continuous function on the interval (0,1). We consider the energy functional $F(u) = \int^1_0\frac{1}{2}((u')^2+u^2)\,dx - \int^1_0fu\,dx$ which is well defined for continuously ...
6
votes
1answer
1k views

Is “Functional Analysis” by “Yosida” a good book for self study?

I was wishing to start studying by myself the book Functional Analysis by Yosida, does anyone have already used it, is it a good reference?