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
0answers
59 views

Compact operators is a linear subspace of bounded operators

Let $X,Y$ be Banach spaces. Let $B(X,Y)$ be the set of bounded linear operators and let $K(X,Y)$ be the set of compact linear operators. I want to prove that $K(X,Y)$ is a vector subspace of ...
0
votes
0answers
30 views

Show $u(x,t)$ is analytic in time

$$u_t + u_x + u u_x - u_{xxt} = 0$$ {know: $u$ can be differentiated $\infty$ times with respect to $t$. this fact may or may not be helpful in the proof} how would one approach such problem? i ...
1
vote
1answer
36 views

Inequality of $L^p$ norm and distribution function

Let $(X,\mu)$ be a measure space and $ f \in L^p$, $ 0 < p < \infty$. Let $ \lambda_f (t) : = \mu ( \{ x \in X : |f(x)| \ge t \} )$. I want to that exists a constant $c_p$ depending only on $p$ ...
2
votes
1answer
33 views

Prove that the space of continuous linear functionals B(X,Y) is complete iff Y is complete

Let $x$ and $Y$ be normed vector spaces and assume that $X\ne\{0\}$. Prove that $B(X,Y)$ - space of continuous linear functionals $A:X\rightarrow Y$ - is complete with respect to the norm ...
1
vote
2answers
57 views

Isometry <=> Adjoint left inverse [duplicate]

Is it true that: $$T\text{ isometric}\iff T^*\text{ left inverse}$$ Obviously: $$\text{"}\Rightarrow\text{": }\langle x,\tilde{x}\rangle=\langle Tx,T\tilde{x}\rangle=\langle x,T^*T\tilde{x}\rangle$$ ...
5
votes
1answer
49 views

Which properties do still hold for the limit of a sequence of functions?

I think it would be very useful to have a list of properties that are preserved for the limit of a sequence of functions, and was wondering if you could help me making the list more complete. Let ...
0
votes
1answer
51 views

Find an analytic function

Is it possible to find an analytic expression for a smooth, continuous, single-variable function $y=f(x)$ such that: 1) $f(0) = 0$ 2) $f(x_1) = y_1$ 3) $f(x) > 0, \forall x>0$ 4) ...
1
vote
0answers
15 views

When are the marginals of an extremal invariant measure also extremal invariant?

Let's suppose that $X$ is a compact metric space, and thus as is $X \times X$. If given a Markov process on $X \times X$ with marginals that are Markov processes on $X$, then we know that the ...
1
vote
1answer
43 views

Check continuity of linear functional and norm

$\mathbb{R}[X] \ni p\rightarrow p' \in \mathbb{R}[X]$ with norm 1) $\|p\|_{\infty}=\sup_{t\in[0,1]}|p(t)|$ 2) $\|p\|_{1}=\int_0^1|p(t)|dt$ 3) $\|p\|_{\infty}^{(1)}=|p(0)|+\sup_{t\in[0,1]}|p'(t)|$ ...
0
votes
1answer
197 views

Prove a non-empty subset is closed in an inner product space

I hope someone would be able to help me with the finer details of this proof. Problem: M is a non-empty set in an Inner Product Space (IPS) X. I need to show that the annihilator of M which is ...
2
votes
0answers
33 views

Bounded linear functionals on $L^\infty$.

I am looking at a practice final and I am a bit confused by this statement I am trying to prove: "There is a nonzero bounded linear functional on $L^\infty[0,1]$ which vanishes on the subspace ...
1
vote
1answer
31 views

Subspaces and convergence in weak* topology

I would like to ask some questions regarding convergence in the weak* topology and subspaces. Let $X$ be a normed space with subspace $A \subset X$. Assume $X$ is endowed with the weak* topology. ...
0
votes
1answer
34 views

Applying perp twice in a hilbert space

Let $H$ be a hilbert space and let $K \subset H$ be a subspace. Then $\overline{K} \subset K^{\perp\perp}$, but why does the reverse inclusion hold?
1
vote
1answer
36 views

Invertible iff Bounded below and dense range

Statement: Given a Hilbert space $\mathscr{H}$ and $\mathscr{K}$ and a bounded operator $A \in \mathscr{B}(\mathscr{H}, \mathscr{K})$. Show that $A$ is invertible if and only if $A$ is bounded below ...
0
votes
1answer
22 views

Show that span is separable [on hold]

Let $X$ be a n.v.s and $A\subset X$. Show that if $A$ is enumerable then $\overline{ \text{span}\{A\}}$ is separable
0
votes
2answers
26 views

Bessel-(LIKE) Inequality

Let $H$ be the Hilbert space, and let $M_1,M_2,...,M_n$ be mutually orthogonal closed linear subspaces of $H$. If $P_{M_i}x=x_i$, then show that $$\sum\limits_{i=1}^n\|x_i\|^2\leq\|x\|^2 ,$$ The ...
1
vote
1answer
35 views

weak-* topologies

Say $S = \{z \in \ell_\infty : z_n \in \{0,1\}\}$. Suppose I am asked a question about the weak-* topology on $S$. How am I supposed to make sense of this? The weak-* topology is a topology on a dual ...
0
votes
0answers
28 views

Banach space and it's compact subsets [duplicate]

I would be very grateful if someone could verify this claim: There are bounded and closed subsets of a Banach space that are not compact.
2
votes
1answer
52 views

How to show: $\exists$ L such that $X\neq L\oplus L^{\perp}$

$X$ : inner product space $L$ : closed subspace of $X$ How to show: $\exists$ L such that $X\neq L\oplus L^\perp$ let $X= (\text{the set of all finite sequences}, \|\cdot\|_2 )$ $L=\lbrace x\in ...
1
vote
1answer
18 views

dist(x,Z)? , Z is span , x(t)=|t|

$Z=$span(sint,cost,sin2t) , x(t)=|t| how to find $dist(x,Z)$ in $L_2(-\pi,\pi)$ dist(x,Z)= $inf ||x-z||$ where $z\in Z$
4
votes
1answer
61 views

Does the open mapping theorem have a local version?

Let $T:X\to Y$ be a linear continuous surjection between Banach spaces $X$ and $Y$. By the open mapping theorem, we have $T$ is open. Now let $C$ be a closed convex subset of $X$ satisfying that ...
1
vote
1answer
27 views

Equicontinuous and pointwise bounded implies compact

Please can you check my proof? I proved: Let $X$ be a compact Hausdorff space and let $C(X)$ denote the space of continuous functions $f: X \to \mathbb R$ endowed with the sup norm $\|\cdot ...
1
vote
1answer
24 views

$\|f\|\le 2$,$|f(x_o)|\le 6$, then $\|x_0\|\le 3$

Let $X$ be a Banach Space, $x_0\in X$. Suppose that for any $f\in X'$ (space of linear bdd functions) with $\|f\|\le 2$ we have $|f(x_o)|\le 6$. How to show: $\|x_0\|\le 3$ could you please help ...
0
votes
2answers
29 views

B-space $X$ is infinite-dimensional implies $X'$ is also infinite-dimensional

We know that the Banach space $X$ is infinite-dimensional, theconclusion we want to show is: then $X'$ is also infinite-dimensional. $X'$: the space of linear bdd functions
3
votes
2answers
180 views

Riesz's Theorem of compactness

$\left(X,\|\cdot\|\right)$ is a normed vector space. $\textbf{Riesz's Theorem of compactness}$ says that $$ \{x \in X \colon \|x\| \leq 1 \} \ \text{compact} \ \Longleftrightarrow \ \text{Each bounded ...
0
votes
1answer
30 views

Norms of linear functionals

I had to check continuity and find norms. I would be very grateful if somebody checked my answers. 1) $\mathbb{R}[X]\ni w \rightarrow w(2) \in \mathbb{R}$ where $\mathbb{R}[X]$ is equipped with norm ...
5
votes
1answer
64 views

Question about Hahn-Banach theorem

Let $(X,\|\cdot\|_1)$ and $(Y,\|\cdot\|_2)$ be normed spaces, and $X\subset Y$. If each $f\in (X,\|\cdot\|_1)^\ast$ extends to a bounded linear functional in $(Y,\|\cdot\|_2)^\ast$ with same norm, ...
2
votes
0answers
42 views
+50

How to prove comparison principle for parabolic PDE (nonlinear)

Suppose $F:\mathbb{R} \to \mathbb{R}$ is smooth with $F(x) > 0$ for $x > 0$ and $F:(0,\infty) \to (0,\infty)$ continuous and increasing with $F(0) = 0$. Consider the PDE $$u_t = \Delta F(u) ...
3
votes
1answer
27 views

Complex exponential is not Fourier multiplier on $L^p$

I am having difficulty to show that the function $m(\xi):= e^{i|\xi|^2}$ is not a Fourier multiplier on $L^p$ when $p\neq 2$. Note that $m:\mathbb{R}^n\to \mathbb{C}$ is called an $L^p$ Fourier ...
2
votes
2answers
54 views

Is this linear functional bounded? Find the norm.

$$\ell^2\ni (x_n)\rightarrow2x_{1}+28x_2+35 x_{3}$$ I think it can be bounded: $$|2x_{1}+28x_2+35 x_{3}| \le |2x_{1}|+|28x_2|+|35 x_{3}| \le 65 (\sum_{n=0}^{\infty}|x_n|^2)^{1/2}$$ But I can't find ...
1
vote
0answers
22 views

Is the $C^r(M, N)$ space, with the strong (Whitney) topology, a Fréchet-Urysohn space?

Given smooth, non-compact manifolds $M$ and $N$, consider the function space $C^r(M, N)$. Equipped with the strong (Whitney) topology, this space is Hausdorff and Baire. It is, however, not first ...
-2
votes
0answers
15 views

side information from equations

Consider system of equations (linear or nonlinear) generated from financial or physical problems. First, the solution to the equations will give information about the problem. Besides the solution, is ...
2
votes
1answer
90 views

Second Derivative of Eigenvalue

Consider the potential $V_t=tx$ and the corresponding Schrödinger operator $H_t=- \frac{\partial ^2}{\partial x^2}+V_t$ on $L^2([0,R])$ with Neumann or Dirichlet boundary conditions. Let ...
-5
votes
0answers
34 views

A separable Hilbert space [closed]

A separable Hilbert space.prove the followong problem:
-2
votes
0answers
46 views

Let $\,E = C([0, 1])$. Set $\,X =(E,||.||_\infty)$ and $\,Y =(E,||.||_1)$. … [closed]

Let $E = C([0, 1])$. Set $\,X =(E,||.||_\infty)$ and $\,Y =(E,||.||_1)$. Let us consider the identity $I :X→Y$. Prove that I is continuous and bijective. Calculate $\,||I||$. Prove that $I^{-1}$ is ...
0
votes
0answers
18 views

Set of limit points of Riemann Integrable functions

I've looked around for answers to this question. It seems like perhaps I don't have enough knowledge of functional analysis to figure out the answer (or even understand the answer), but I'm intrigued. ...
-4
votes
0answers
23 views

let E=C[X] be a normed space and T∈ L(E)… prove that.. [closed]

Let E=C[X] be a normed space and T∈ L(E). And let $$\||P||_\ = \left\{ \sum ||P^{(n)}||_\infty, \; \; 0 \leq n \leq ∞ \right\}.$$ where $\||P||_\infty$=sup|p(x)|, 0≤x≤1 1- Justify that T:E→E ...
1
vote
1answer
23 views

Show compactness of an operator with Arzelà–Ascoli

We have $K\colon L^{2}(a,b) \rightarrow L^{2}(a,b)$ such that $ Kf(t)=\sum_{j=1}^{n}\phi_{j}(t) \int_{a}^{b} \psi_{j}(S) f(s)ds$ where $\phi_{j} ,\psi_{j} \in L^{2}(a,b)$. We want to show that K is ...
3
votes
2answers
106 views

Proof of equicontinuous and pointwise bounded implies compact

I tried to prove the Arzela-Ascoli theorem: Let $X$ be a compact Hausdorff space and let $C(X)$ denote the space of continuous functions $f: X \to \mathbb R$ endowed with the sup norm $\|\cdot ...
2
votes
1answer
53 views

is this true for Hilbert space direct sum of $H$ when $H$ is infinite dimensional?

Let $(H_{\alpha})_{{\alpha \in I}}$ be a $I-$indexed family of Hilbert spaces over $\mathbb{F}$. let $H=\bigoplus H_\alpha$ be their Hilbert space direct sum. Can we say $\dim ...
0
votes
1answer
35 views

Prove that $F$ is dense in $C(X\times Y,\mathbb{R})$?

Let $X$ and $Y$ be compact metric spaces. Let $$ F= \Bigl\{\sum_{i=1}^n A_i f_i(x) g_i(y): f_i\in C(X,\mathbb{R}),g_i\in C(Y,\mathbb{R}), 1\le i\le n \Bigr\}. $$ Prove that $F$ is dense in $C(X\times ...
0
votes
1answer
17 views

Nonexpansive Affine Operators in Hilbert spaces

Let $H$ be a Hilbert space with real inner product $\langle \cdot, \cdot \rangle : H \times H \rightarrow \mathbb{R}$. Let $T$ be an affine operator. Show that $T$ is nonexpansive, i.e., $\left\| ...
1
vote
1answer
44 views

Notation question A $\subset \subset B$

I am a bit confused about the notation A $\subset \subset B$ used in functional analysis. The definition I have says: $A \subset \subset B$ iff $A \subseteq B$ and $\bar{A}$ compact in $B$. Wikipedia ...
0
votes
0answers
36 views

Prove that B is bounded ?? [closed]

Let G be a Banach space and let B be a subset of G. Suppose that f∈G* we have f(B) = {f(x); x∈B} is bounded in R. Prove that B is bounded. Such that G* is the dual space.
-1
votes
0answers
19 views

prove that F is dense in C(X×Y,R) ? any help! [closed]

Let $X,Y$ be compact metric spaces. Let $$F= \left\{ \sum A_i f_i(x) g_i(y), \; f_i \in C(X,\mathbb{R}), \; g_i \in C(Y,\mathbb{R}), \; 1 \leq i \leq n \right\}.$$ Prove that $F$ is dense in $C(X ...
1
vote
0answers
26 views

Fourier transform of $e^{it\sqrt{a^2+x^2}}$

The question is clear, I came up with this Fourier transform to calculate while searching explicit solutions for a PDE, but I don't even know if it is feasible. $ \mathcal{F}_x(e^{it\sqrt{a^2+x^2}}) ...
1
vote
1answer
51 views

Norm of linear functional with e

I have to find norm of this functional: $$\ell^{4/3} \ni (x_n)_1^{\infty} \rightarrow \left(\left(1+\frac{1}{n}\right)^n x_n\right)_1^{\infty} \in \ell^{4/3}$$ I proved that this functional is ...
0
votes
1answer
59 views

Theorem with an example

I have this theorem In the paper they give an example: But here $H_1$ is not satisfied ! How to correct it please? http://mathoverflow.net/questions/163788/theorem-with-an-example
1
vote
1answer
19 views

An unproved statement from Gohberg and Krein

In the book: Introduction to the Theory of Linear Nonselfadjoint Operators in Hilbert Space, by Gohberg and Krein, page 3, there is an unproved statement that seems to be pulled from nowhere. Let P ...
-1
votes
1answer
55 views

norm of integral operator in $C([0,1])$

If we define on $C([0,1])$ the operator $$ Tf(x) = \int_{0}^{1} K(t,s) f(s) ds$$ where $K$ is a continous function on two variables. I want to show that: $1)$ $||T|| = \displaystyle\max_{t} ...