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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1answer
65 views

Geometrical representation of the unit ball?

Let $E$ be the vector space of $\mathbb{R}$-valued continuous functions on $[0\ 1]$. With the norm $\| f \| = \max \{\ | f (x) |; 0 \leq x \leq 1\}$, the open ball centered at $f$ and radius $r$ has ...
1
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1answer
54 views

Prove a function has a maximum and minimum along a domain

Given the function $f:[13,132] \to R$ defined by $f(x)=sinx+x^3-$2 $e^x $ prove that the function has a maximum and minimum along the domain. I understand that a function has a maximum and minimum ...
2
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0answers
28 views

positive elements in $L(H)$

At the moment I'm studying positive elements in $C^*$-algebras. Let $H$ be a complex Hilbert space, $L(H)$ the linear bounded operators $H\to H$ and $T^*=T$. Then: $$\sigma(T)\subseteq [0,\|T\| ]\; ...
0
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0answers
21 views

Need simple logic or formula for the below problem!

The problem is simple tip calculator here calculating remaining tip from the money got from user. Inputs - x,y,z Where "x,y" are two denominations of currency and "z" is billamount If x = 2, y=5, ...
0
votes
1answer
17 views

Dual space of a finite dimensional

Let $V$ be a normed space with dual $V^*$. Then $V$ is finite dimensional if and only if $V^*$ is finite dimensional, and in fact $\dim{V} =\dim{V^*}$ I set up the proof as follows: since $V$ is ...
0
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0answers
21 views

Does any continuous map between two Banach spaces over a nonarchimedian field is a closed map?

Does any continuous map between two Banach spaces is a closed map? It seems to me that it is true. Let $f:V \rightarrow W$ be a map, where $V$, $W$ are (infinite dimensional) Banach space over a a ...
2
votes
1answer
24 views

Dual space of a finite dimensional is finite dimensional

Let $V$ be a normed space with dual $V^*$. Then $E$ is finite dimensional if and only if $V^*$ is finite dimensional, and in fact $\dim{V} =\dim{V^*}$. I set up the proof as follows: Let ...
1
vote
0answers
25 views

How to prove $f$ is outer, when $Re f$ >0?

This is an exercise problem, which can be found in most of the book which deals with Hardy Space. Any kind of suggestion or hint is also welcome. The question is following: Let $f$ be a function in ...
2
votes
2answers
101 views

Prove that $T(B)$ is relatively compact in $C([a,b])$.

Let $B$ be the unit ball in $C([a,b])$. Define for $f\in C([a,b])$, $$Tf(x)=\int_a^b (-x^2+e^{-x^2+y})f(y)dy.$$ Prove that $T(B)$ is relatively compact in $C([a,b])$. My attempt: If $|f(x)| \le ...
0
votes
1answer
42 views

Show that the functional is continuous everywhere in $V$

Let $J: V \to \mathbb{R}$ be a linear functional and $V$ a linear space with norm. Show that if $J$ is continuous on $0 \in V$ then $J$ is continuous everywhere in $V$. That's what I have tried: ...
2
votes
2answers
44 views

Intersection of Hilbert spaces

Consider two Hilbert spaces $H_1$ and $H_2$ with inner products $\langle \cdot,\cdot\rangle_1$ and $\langle \cdot,\cdot\rangle_2$ generating norms $\Vert \cdot \Vert_1$ and $\Vert \cdot \Vert_2$ ...
4
votes
2answers
102 views

Compute Quotient Space

I have been struggling with this computation for a while now. I thought I was almost there, but it now results I still have nothing. So here is the initial problem: Let $c=\left\{ (x_j)_j \subset ...
2
votes
0answers
34 views

independent symmetric 3-valued random variables in Lp

Consider the following excerpt from this paper: Given $1<p<2$, $0<w\leq 1$ and $n\in\mathbb{N}$, we fix once and for all a sequence $f_j^{(n)}=f_{p,w,j}^{(n)}$, $1\leq j\leq n$, of ...
0
votes
1answer
28 views

continuous functional calculus for nonunital $c^*$-algebras

In lecture we had the continuous functional calculus for unital $c^*$-algebras: Let $A$ be an unital $c^*$-algebra, $a\in A$ normal and let $$alg(a,a^*)=\overline{ \{ ...
1
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1answer
19 views

Theorem 3.3-1, Lemma 3.3-2, and Theorem 3.3-4 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS: How to write these as one?

I'm trying to prepare some ancilliary material on the following three results in sec. 3.3 in the book Introductory Functional Analysis With Applications by Erwine Kreyszig: (First, I'm giving ...
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vote
2answers
39 views

Norm inequality

While trying to compute a quotient space, the next problem has come to my attention: Let $x=(x_j)_j$ and $y=(y_j)_j$ be two complex convergent sequences such that $x-y=(x_j-y_j)_j$, is a constant ...
1
vote
1answer
24 views

Degree of map on $U(n)$ and roots in $U(n)$

Recently I went to a talk of A.Thom in which he sketched a proof of the fact that the groups U(n) satisfy the Kervaire-Laudenbach conjecture. At some point in the proof you have to argue that the map ...
1
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1answer
34 views

Eigenvector of a $C^n$ class matrix

Let $A$ be the following matrix function: $\Bbb{R} \to \Bbb{R}^{a \times (a+1)}$ $t \mapsto A(t)$ Let us suppose that $A$ is $C^{\infty}$, meaning that all of $A$ coefficients are $C^{\infty}$. Let ...
2
votes
5answers
40 views

Equivalent Norms $\|x\|_1=\|x\|+|f(x)|$

Let $f: X\to \mathbb{R}$ be a linear functional and $\|\cdot\|_1$ is defined as follow $\|x\|_1=\|x\|+|f(x)|$. Prove or disprove $\|\cdot\|_1$ is equivalent to $\|\cdot\|$ iff $f$ is continuous. I've ...
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0answers
48 views

functional inequality with a strange constant at RHS

I want to prove the following result. Let consider a function $f$ twice continuously differentiable from $[0,1]$ into $\mathbb{R}$ such that ...
2
votes
0answers
29 views

Hamiltonian: Invariant Core

Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ And an operator: $$A:\mathcal{D}(A)\to\mathcal{H}:\quad A=A^*$$ Denote its evolution by: ...
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vote
1answer
18 views

Reducing Subspaces: Hamiltonian

Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Regard a projection: $$P\in\mathcal{B}(\mathcal{H}):\quad P^2=P=P^*$$ Then one has: ...
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1answer
30 views

Quotient space of infinite dimensional vector space

On an exam today I used that if $X=\mathcal{C}[a,b]$ and $Y=\{f\in X : f(a)=f(b)\}$, then the projection $\pi: X\rightarrow X/Y$ has the property $\ker(\pi)=Y$. This led me to the following: Suppose ...
2
votes
0answers
23 views

weakly open subset in $M[0,1]$ (the space of finite measures on $[0,1]$)

I came across this question when reading Lynch and Sethuraman (1987): Large deviations for processes with independent increments. Let $M[0,1]$ be the space of finite nonnegative measures on $[0,1]$ ...
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0answers
34 views

Compactification of Polish space , What is its use?

I want to know the use of the fact that a Polish space can be homeomorphically embedded into a dense subset of a compact metric space. For example, a continuous function $f$ on a Polish space can't ...
2
votes
1answer
25 views

Show that $C_{0} = \{(a_n)_{n \in \mathbb{N}}\subset \mathbb{R}: a_n \rightarrow 0\}$ is complete.

Show that $C_{0} = \{(a_n)_{n \in \mathbb{N}}\subset \mathbb{R}: a_n \rightarrow 0\}$ is complete. I've already seen that this question has been asked, and already answered, however, I've managed ...
0
votes
0answers
32 views

Idea of the proof of Lusternik-Schnirelmann

I have this theorem of Lusternik-Schnirelmann from Chang's book: " Let $M$ be a smooth Banach-Finsler manifold. Suppose that $f\in C^1(M,\mathbb{R})$ is a function bounded from below, satisfying ...
0
votes
2answers
31 views

Periodic function

Considering the function $F(x)=x-E(x)$ such as $E(x)$ is the integer part of $x$ So here just with observation we can see that : $f(x+1)=(x+1)-E(x+1)=f(x)$. But here mathematics is not based just ...
1
vote
1answer
19 views

Selfadjoint Operators: Weak Convergence

Given a Hilbert space $\mathcal{H}$. Consider a selfadjoint operator: $$A:\mathcal{D}(A)\to\mathcal{H}:\quad A=A^*$$ Regard a sequence: ...
0
votes
1answer
34 views

Laplace transform, Bochner integral

I have a quesition about linear operators on a Banach space. Let $B$ be a real Banach space. $(T_{t})_{t>0}$ is called strongly continuous contraction semigroup on $B$ if For all $t>0$, ...
0
votes
0answers
18 views

Integral kernels of self-adjoint operators

If the integral kernel $k(x, y)$ of an operator $T : C^\infty_c(M) \to \mathcal{D}'(M)$ is symmetric ($M$ is a compact manifold), then the operator $T$ is symmetric. Is the converse true? That is, ...
1
vote
1answer
51 views

A copy of $l_\infty$ in a infinite dimensional Banach space

Let $E$ an infinite dimensional Banach space. Using the Hahn-Banach extension theorem, prove that there is a sequence $(y_n)\subset E$ and a decreasing sequence of closed subspaces ...
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vote
0answers
66 views

Existing complete function space allowing discontinuity .

This is a question which came to me due to several previous question: sorry for the all previous links necessary to look to get the question. The latest question is in the link: Convergence on Norm ...
1
vote
0answers
40 views

What's the maximum speed of snake so that the frog can escape?

Suppose there's a round pond, a frog which can swim as 1 meter / second, and a snake that moves along the pond ridge but cannot swim. If the frog can reach any point on the ridge of the pond before ...
0
votes
1answer
38 views

conditions for norm of linear bounded operator to satisfy $\lvert T_x (y) \rvert = \lVert T_x \rVert$.

Let $x = (x_n)_{n \in \mathbb N} \in l^\infty$ and let $T_x : l^1 \rightarrow \mathbb F$ be defined by $T_x (y) = \sum_{n=1}^\infty x_ny_n$. What condition on $x$ is needed so that there exists $y \in ...
4
votes
0answers
73 views

Convergence on Norm vector space.

I am not sure if this question make sense mathematically, so please bear with my ignorance. This is an extension to the question in the link: Is complete metric space is required? It seems in many ...
3
votes
2answers
35 views

Show $ \langle Tx,x \rangle \in \mathbb R$ for all $x \in H$ implies $T$ is self-adjoint

Show that a linear operator $T: H \rightarrow H$ is self adjoint if and only if $\langle Tx, x \rangle \in \mathbb R$ for all $x \in H$. You may use that the equality that for all $x,y \in H$ ...
1
vote
1answer
64 views

How do I show convergence of this sequence?

Given a sequence $\{a_n\}$ of positive real numbers such that $\sum\limits_{n=1}^\infty a_n<\infty$. Suppose that there exists $k\in \Bbb N$ such that $a_{n+k}\leq a_n, \,\,\,\forall n.$ Question: ...
9
votes
1answer
171 views

Growth of Tychonov's Counterexample for Heat Equation Uniqueness

Define a function $\varphi$ on $\mathbb{R}_{+}$ by $$\varphi(t):=\begin{cases}e^{-1/t^{2}}, & {t>0}\\ 0, & {t\leq 0}\end{cases}\tag{1}$$ It is well-known that $\varphi$ is ...
1
vote
1answer
42 views

Is complete metric space is required?

This question may be quite related to the following link: but I am not sure. Sorry, if it is trivial. Advantage/disadvantage of complete/incomplete metric space. In many application specially in ...
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0answers
31 views

Advantage/disadvantage of complete/incomplete metric space.

It must be simple. I understand a metric space can be complete for a given metric and and the same set may be incomplete with a different metric. This may be due to the fact that under the given ...
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votes
0answers
20 views

Necessity of continuity in Topological Vector Space

In the notion of a topological vector space, we define such as a vector space $X$ (over a field $\mathbb{K}$) with topology $\mathscr{T}$ such that $$\iota_+: (X,\mathscr{T}) \times (X, \mathscr{T}) ...
2
votes
0answers
22 views

Trick to rewrite operator in terms of another?

In the book Beautiful Models: 70 Years of Exactly Solved Quantum Many-Body Problems by Bill Sutherland, I would like to understand the trick done in (4), see the excerpt from p 29 shown below I ...
0
votes
1answer
27 views

Bounded linear functionals over smooth maps of a compact interval

I have two questions regarding the topological dual of the space $E = \mathcal{C}^\infty([0; 1])$ of infinitely continuously differentiable functions over the closed interval $[0; 1]$ equipped with ...
6
votes
0answers
71 views

What is the strongest possible statement of the idea that “the tangent line is the best linear approximation”?

For instance, I've just checked that that if you take the best linear approximation (in the $L^2$ sense) to a sufficiently nice function $f$ on the interval $[-\varepsilon, \varepsilon]$, and then let ...
1
vote
1answer
21 views

Net converging weak-*, implies uniform bound?

Let $E$ be a complex Banach space. A consequence of the uniform boundedness principle is the following. If $(\lambda_n)_{n\geq 1}$, $\lambda$, are elements of $E^*$ such that $$ \lambda_n(x) ...
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0answers
22 views

Is there a pseudocontractive mapping that is not strictly pseudocontractive?

Given a Hilbert space $H$, a mapping $T:H\rightarrow H$ is said to be pseudocontractive if $$\|Tx-Ty\|^2\leq \|x-y\|^2+\|(x-Tx)-(y-Ty)\|^2\,\,\, \forall x,y\in H,$$ and it is strictly ...
1
vote
1answer
21 views

Show that a subset of $(\mathbb R^n,||.||)$ is closed

Let $C$ be a closed subspace of the normed linear space $(\mathbb R^n,\| \cdot \|)$.Let $r(>0)\in \mathbb R$ Define $D:=\{y:\exists x\in C$ such that $\|x-y\|=r\}$. Show that $D$ is closed. My ...
2
votes
1answer
56 views

Showing this function is continuous $ f:(x,y)\mapsto x^2+y^2$

I have the following function: $$f:\Bbb R^2 \to \Bbb R,\quad f:(x,y)\mapsto x^2+y^2$$ I want to show that this function is continuous by showing that $f^{-1}((a,b))$ is an open set. How do I ...
4
votes
1answer
56 views

How to do it by Dominated Conversgence Theorem?

I'm trying to find the limit $$ I = \lim_{n\to\infty} \int_{\mathbb R^d} \frac1{n} |f(x)|^2 x\cdot\nabla\chi (x/n)dx, $$ where $f \in H^1 (\mathbb R^d, \mathbb C)$, $f \in H^2_{loc}(\mathbb R^d, ...