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)

1
vote
3answers
330 views

Counterexample for Mazur–Ulam theorem

We know, Mazur–Ulam theorem states that if $V$ and $W$ are normed spaces over $\mathbb{R}$ and the mapping $f\colon V\to W$ is a surjective isometry, then $f$ is affine. Can somebody say ...
1
vote
3answers
97 views

Finding spectrum of the operator A

$A:\mathcal{l}_2\rightarrow \mathcal{l}_2:(x_n)_{n=1}^\infty \rightarrow (x_{n+1})_{n=1}^\infty$ (left shift) Find the spectrum and all its parts for the operator A. What should I do?
1
vote
1answer
59 views

Hilbert space inequality $|\langle x,y\rangle | \leq \epsilon ||x||^2 + C_{\epsilon}||y||^2$

In prelim prep I came across 'given $\epsilon$ there exists $C_{\epsilon}$ such that $|\langle x,y\rangle | \leq \epsilon ||x||^2 + C_{\epsilon}||y||^2$. It is asserted without proof, so I've tried ...
1
vote
1answer
59 views

Some Help to prove $\|T^{-1}\|=\|T\|^{-1}$

I'm trying to prove that given to Banach spaces $X,Y$, and a continuous linear transformation $T:X\to Y$ with bounded inverse $T^{-1}:Y\to X$. Then, $$\|T^{-1}\|=\|T\|^{-1}$$ I already know that ...
1
vote
2answers
98 views

Euclidean norm injective?

I seem to have thought myself into a corner. Can someone point out the hole in my reasoning here. Suppose $f:X \rightarrow \mathbb R$ where $f(x) = \|x\|,$ for $x$ in $X$. Knowing that $\|x\|=0$ ...
1
vote
1answer
230 views

Example of a separable space without a Schauder basis.

Can I say that the normed linear space $(\Bbb{R}(\Bbb{Q}), \lvert\, \cdot\,\rvert)$ is an infinite dimensional, separable, Banach space and hence cannot have a Schauder basis? My argument is based on ...
1
vote
3answers
154 views

Can $\le$ be used insted of < in the definition of continuity?

A common definition of a continuous map $T:M_1\to M_2$ is that for every $x\in M_1$ and every $\epsilon>0$ there exists a $\delta >0$ such that for all $y$ in $M_1$ $$d_1(x,y)<\delta \implies ...
1
vote
2answers
147 views

Applying the Banach's Contraction Principle

I have a problem: For a system of linear equations: $$x_i=\sum_{j=1}^{n}a_{ij}x_j+b_i,\ \ i=1,2, \ldots , n \tag 1$$ Prove that, if $$\sum_{i=1}^{n}\sum_{j=1}^{n}a_{ij}^2 \le q<1$$ then ...
1
vote
1answer
124 views

Bounded Self-adjoint Operator on Hilbert Space

I am trying to show that if $A$ is a bounded, self-adjoint and positive operator on a Hilbert space $H$, $0 \in \rho(A)$, the following inequality holds for all $x \in H$ with $\|x\| = 1$: ...
1
vote
2answers
62 views

A question about a linear bounded operator (in hilbert space)

I am reading a book about C*-algebra. When i study von Neumann algebras in this book, i meet with a problem. In the book, If $H$ is a Hilbert space, we write $H^{(n)}$ for the orthogonal sum of n ...
1
vote
3answers
169 views

The closed form of of the Dirac Delta Function

I know that $\delta(x)$ is the Dirac Delta function - satisfying -$$\int^{\infty}_{- \infty}\delta(x)dx=1$$Can anyone tell me a closed form of $\delta(x)$ . I guess it might be in a form of ...
1
vote
1answer
219 views

Comprehensive references on partial differential equations

How do the three volumes by Taylor's "Partial differential equations" compare with the two volumes with the same title by Friedrich Sauvigny's as a reference for study? What are the good and bad ...
1
vote
1answer
84 views

Is a set of single element $\{x\}$ connected in a metric space $(X,d)$?

Is a set of single element $\{x\}$ connected in a metric space $(X,d)$? Definition: Suppose that $(X,d)$ is a metric space. A set $E \subseteq X$ is said to be disconnected if there exist two ...
1
vote
2answers
509 views

$P_n[0,1]$ be the set of all polynomial of degree atmost $n$ with supnorm is it a closed in $C[0,1]$? [duplicate]

$P_n[0,1]$ be the set of all polynomial of degree atmost $n$ with supnorm is it a closed in $C[0,1]$? and $P[0,1]$ is set of all polynomials in $C[0,1]$ I know which is dense in $C[0,1]$ as we know ...
1
vote
2answers
89 views

Relationship between two projectors

Please, somebody can help me with this problem? Let $V$ and $W$ be two closed subspaces of a Hilbert $(H, \langle \cdot,\cdot\rangle)$, and let $P:H\rightarrow V$ and $Q:H\rightarrow W$ the ...
1
vote
3answers
141 views

$\|f*g\|_q\leq \|g\|_q \|f\|_1$ and $\|f*g\|_\infty\leq \|g\|_q \|f\|_{q^{'}}$, $(1/q+1/q^{'}=1)$?

Now I'm reading the Young inequality. It says that if $f \in L^p(R)$, $g \in L^q(R)$, $1\leq p,q\leq \infty$, $1/p+1/q\geq 1$. Then how could we have the following inequalities: $$\|f*g\|_q\leq ...
1
vote
3answers
104 views

Convergence in $L^1$

Let $f \in L^1(R)$. Show that $\sum^{\infty}_{n=1} f(x + n)$ converges a.e. Solution: So, ultimately we are going to want $\sum^{\infty}_{n=1} f(x + n) \leq$ something in $L^1$ that converges ...
1
vote
1answer
95 views

$l^{p}$ is not finite dimensional

Well, the exercise was to prove that $l^{p}$ is not finite dimensional space for $p$=2. I did it proving that the unit ball is not compact. Easy. However, i was trying to build an element $x \in ...
1
vote
2answers
77 views

Question on continuity of functions from $X\times X\rightarrow Y$

I am stuck on the following problem, which I do not believe to be so difficult. Let $X$ and $Y$ be Banach spaces. Let $f:X\times X\rightarrow Y$ be a function such that for any fixed $x_0$, ...
1
vote
1answer
115 views

$ \int_{-\infty}^\infty |f| < \infty$. Then $ \lim_{x \to \infty} f(x) =0\;?$ [duplicate]

Possible Duplicate: If $\int_0^\infty fdx$ exists, does $\lim_{x\to\infty}f(x)=0$? Let $ \int_{-\infty}^\infty |f| < \infty$. Then $$ \lim_{x \to \infty} f(x) =0 \;?$$ If this is true, ...
1
vote
2answers
138 views

Find the error in proof that $\|x\|_{\ell^1} \le \|x\|_{\ell^2}$

For sure, $\ell^2$ is larger than $\ell^1$, because for $|x|<1$, $|x|^2<|x|,$ that is, $||x||_2\leq||x||_1.$ But using Cauchy-Schwarz inequality, I get a "wrong" comparison: ...
1
vote
1answer
94 views

Proving convexity of this set in $\ell^2$

This is a follow-up to the question I posted earlier this week. Consider, for a fixed sequence $(a_n)_n\in\ell^2$ the subspace $$C=\{(x_n)_{n}\in\ell^2 : |x_n|\le a_n\text{ for all ...
1
vote
2answers
103 views

Is it true that $ T(\overline{B(0,1)}) = \overline{T(B(0,1))} $ for continuous operator $T$?

$T : X \to Y$ be a surjective linear continuous operator($X, Y$ : Banach Space). Then does $$ T(\overline{B(0,1)}) = \overline{T(B(0,1))} $$ hold? Here $B(0,1)$ means an open ball in $X$, and Bar ...
1
vote
2answers
66 views

For any sequence in $L^2$ there is a function in $L^2$ s.t. is not orthogonal to any point of the sequence

How to prove that for any sequence $(f_n) \subset L^2[0,1]\setminus \{0\}$ there is a function $g \in L^2[0,1]$ such that $$\int f_n g dx \neq0\ \forall n\geq 1?$$ I tried to use a weak limit of ...
1
vote
1answer
80 views

Show that $(a_1,a_2/2,a_3/3,a_4/4,\ldots)$ is not dense in $\ell_\infty$

Let $T\colon \ell_\infty \rightarrow \ell_\infty : (a_1,a_2,\ldots) \mapsto (a_1,a_2/2,a_3/3,a_4/4,\ldots) $. Show that $\operatorname{range}(T)$ is not dense in $\ell_\infty$. I want to ask for ...
1
vote
2answers
388 views

What is the topological dual of $C_b(\mathbb{R})$

Consider the Banach space $C_b(\mathbb{R})$ of continuous bounded functions on $\mathbb{R}$ equipped with the sup-norm. 1) Do we know a precise description of its topological dual ...
1
vote
2answers
730 views

How do the solutions to the wave and heat equations converge in general?

I would like to check my understanding with someone if possible. When we cover the heat and wave equations, for instance, in "methods" courses at university, they normally restrict the initial ...
1
vote
2answers
720 views

Example of analytic piecewise-defined function

Does there exist an analytic everywhere, piecewise-defined function $f$ such that: $f(x) = g(x)$ for $x < k$ $f(x) = h(x)$ for $x>k$ $f(x) = r$ for $x=k$ With $g \ne h $ ($g$ not the same ...
1
vote
3answers
992 views

Is a function analytical on C iff its Fourier-transform vanishes for negative frequencies?

I think Cauchy's integral formula and the Hilbert transform can be used to prove one direction, but is this an equivalence or only an implication? edit for clarification: Is a function $f : \mathbb C ...
1
vote
1answer
20 views

Bound on the product of functions in $L^1$

Let X be a bounded subset of $\mathbb{R}$ and let $f, g,$ and $h$ be real valued functions in $L^2(X)$. Consider $$\| fgh\|_{L^1(X)}.$$ The hope is to get an upper bound in terms of ...
1
vote
3answers
66 views

Holder Continuous Functions on [0,1] are complete + Banach space

I am studying for an Analysis prelim and I am stuck on how to show that the following space of Holder continuous functions is complete: $$\Lambda_{\alpha}([0,1]) = \{f: [0,1] \rightarrow \mathbb{R} ...
1
vote
2answers
45 views

$\ker f$ is either dense or closed when $f$ is a linear functional on a normed linear space

Let $f$ be a linear functional on a normed linear space $X$. Prove that $\ker f$ is either dense or closed in $X$. Two possibilities can occur, i.e either $f$ is bounded or unbounded. If it is ...
1
vote
2answers
51 views

Can I define a bounded sequence whose Banach limit is not unique?

Banach limit, as a non-constructive object, is not unique. The Banach limit for some sequences, say, convergent sequences, sequences satisfying $a_n = a_{n+m}$ for all $n$ and some $m$, the Banach ...
1
vote
2answers
32 views

How can I prove this operator is not continuos

Let $X$ the normed space of all polynomials on $J=[0,1]$ such that $||x||$=max$|x(t)|$ $t \in[0,1]$ and we have the following operator $Tx(t)=x'(t)$ prove this operator is not continuous
1
vote
1answer
34 views

Exercise of comager set and the space $C([0,1])$

Let $X$ be a topological space and $P \subseteq X$ holds generic. If $P$ is comeager , we say that $P$ hold generically or that the generic element of $X$ is in $P$ Show that the generic element of ...
1
vote
1answer
81 views

Hilbert space and Parallelogram law

Let $ C_\infty$ be inner product space of all real sequences $\{x_n\}$ with $x_n$ finite number of nonzero terms and the inner product defined by $$\langle x,y\rangle =\sum_{i=0}^\infty x_ny_n$$ I ...
1
vote
1answer
50 views

Is $C[0,1]$ locally Compact?

I'm asked to use the function $f_n(x)=nx$ for $0\le x\le \frac{1}{n}$ and $f_n(x)=1$ for $\frac{1}{n}\le x\le 1$. I'm not familiar with Functional Analysis.
1
vote
1answer
39 views

embedding of $\prod_{n\in\mathbb{N}}M_{n}(\mathbb{C})$ in a type $II_{1}$ factor

Suppose $M$ is a type $II_{1}$ factor with trace $\tau$. Let $\lbrace p_{n}\rbrace_{n\in\mathbb{N}}$ be an increasing sequence of projections such that $\tau(p_{n})\rightarrow 1$. Now, let's consider ...
1
vote
1answer
62 views

Does weak convergence in $L^{q}$ imply weak convergence in $L^{p}$

Assume we have $u_{k} \rightharpoonup u$ in $L^{q}(\Omega)$, does it then follow that $u_{k} \rightharpoonup u$ in $L^{p}(\Omega)$, given that $q > p$ and $\Omega \subset \mathbb{R}^{n}$ is ...
1
vote
3answers
79 views

Equivalency of Norms and the Open Mapping Theorem

"Let $\|\cdot\|_1$ and $\|\cdot\|_2$ be two norms on the space $X$ s.t. $(X,\|\cdot\|_1)$ and $(X,\|\cdot\|_2)$ are both complete. Assume that for any sequence $(x_n) \subseteq X$, $\|x_n\|_1 \to 0$ ...
1
vote
1answer
145 views

An abelian Banach algebra without characters

Can one give an example of an abelian Banach algebra with empty character space? Such algebra must be necessarily non-unital. I couldn't find any examples of such algebras. Thanks!
1
vote
1answer
88 views

Example of a proper, dense ideal in a unital Banach algebra

I know that if ${\cal A}$ is a non- unital Banach algebra, then ${\cal A}$ may contain some proper, dense ideals. I need an example of a proper, dense ideal in a unital Banach algebra or show that it ...
1
vote
1answer
72 views

Continuous spectral theorem example

The spectral theorem can be explicitly expressed for an hermitian matrix by providing its eigen decomposition. In the more general case of a bounded self-adjoint operator with a continuous spectrum, ...
1
vote
3answers
105 views

Showing $T: X\rightarrow Y$ is a linear map, is one-to-one… Over-thinking question?

so my question is as follows: Suppose that $X$ and $Y$ are normed linear spaces and that $T: X\rightarrow Y$ is a linear map (ie $T(\alpha x_1+\beta x_2) = \alpha T(x_1) + \beta T(x_2) \forall ...
1
vote
1answer
263 views

Extreme points of the unit ball of $l^1(\mathbb{Z^+})$ and $L^1[0,1]$

Determine the extreme points of the unit ball of $l^1(\mathbb{Z^+})$ and $L^1[0,1]$. My attempt: I know the definition but I don't know how to find these extreme points.Please help me to solve this ...
1
vote
2answers
67 views

$2\pi$ in the Definition of Fourier Transform

Most textbooks I read define Fourier transform of a function $f \in L^2(\mathbb R)$ as $$ \hat f (\xi) := \int_\mathbb R f(x) e^{-2\pi i x \xi} dx. $$ However, in class my teacher defines it without ...
1
vote
4answers
114 views

Completion of pre hilbert space

Let $L$ be a Hilbert space and $T$ be a linear densely defined operator, $T: D(T)\subset L \to L$ , $\overline {D(T)}=L$ We can make $D(T)$ , a prehilbert space by defining an inner product $\langle ...
1
vote
1answer
40 views

Does $\|a\|^2 \le \|a^\ast a\|$ imply $\|a^\ast \| \|a\| \le \|a^2\|$?

Let $A$ be a normed algebra with an involution $\ast$. As per title, my question is whether $\|a\| \|a\| \le \|a^\ast a\|$ implies $\|a^\ast \| \|a\| \le \|a^2\|$. I used it in a proof I wrote ...
1
vote
1answer
45 views

If $u_n^p \rightharpoonup v$ in $L^1$, then does it follow that $u_n \rightharpoonup v^{\frac 1p}$ in $L^p$?

Let $\Omega$ be a bounded domain. Suppose that $u_n^p \rightharpoonup v$ in $L^1(\Omega)$. Does it follow that $u_n \rightharpoonup v^{\frac 1p}$ in $L^p(\Omega)$?
1
vote
1answer
83 views

Hahn Banach and separation of points

I understand why, given Hahn-Banach, for some nonzero $x\in X$ there exists a $f\in X'$ such that $||f||=1$ and $f(x)=||x||$. But why is it also so there exist $f\in X'$ such that if $f(x)=f(y)$ this ...