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Questions tagged [functional-analysis]

Functional analysis, 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 and other topics. For basic questions about functions use more suitable tags like (functions), (functional-equations) or (elementary-set-theory).

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, including spectral theory, as well as measure, integration, probability on infinite dimensions, and also manifolds with local structure modeled by these vector spaces.

For basic questions about functions use more suitable tags like , or .

52582 questions
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Uniform closure of step maps

Let $E$ be a Banach space. Define step map $f:[a,b]\to E$ as a map for which there is a partition $a=a_0, a_1, \ldots, a_n = b$ and elements $v_1,\ldots, v_n\in E$ such that $f(t)=v_i$ for $t\in(a_{i-1},a_i)$. $a,b,E$ are fixed. What is the closure…
Canis Lupus
  • 2,565
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Show that {$x \in l^\infty: \sum_{n=1}^\infty x_n=0 \land x_n=0$ for almost all n $\in \mathbb N$} dense in $l^2$

Let $A := \{x \in \ell^\infty: \sum_{n=1}^\infty x_n=0 \land x_n=0$ for almost all $n \in \mathbb N \}$. How can you prove that $A$ is dense in $\ell^2$ and not in $\ell^1$? And what is $A^\bot$ in $\ell^2$?
red
  • 327
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cokernel and kernel of adjoint operator

Let $L$ be a linear operator from Banach space $X$ to $Y$. Is the dimension of the kernel of the adjoint of $L$ the same as the dimension of the cokernel? The cokernel is $Y/(Im L)$. Also, is the index of operators for which this quantity is…
Q-the-curvature
  • 517
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existence of eigenfunction for an operator

I'd like to know whether there's a general condition on an operator for it to have an eigenfunction. For example, differential operator has eigenfunction $f_k (x)=e^{kx}$ , and differential operator have many properties such as linear and obeys the…
iridium
  • 395
2
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1 answer

Show that the limit is zero

I have to show that if $(x_n)$ is weakly convergent in $X$ then for any $a>1$ $$\lim_{n\rightarrow\infty}\frac{\|x_1+\dots + x_n\|}{n^a}=0$$ My attempt: If $(x_n)$ is weakly convergent, then it is bounded by some constant $C>0$. So we…
luka5z
  • 6,359
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2 answers

If $X,Y$ are Banach spaces then for every $(y_n)\rightarrow 0 \exists (x_n)\rightarrow 0 : f(x_n)=y_n$

Let $X,Y$ be Banach spaces and $f:X \rightarrow Y$ be a linear, continuous and surjective map. Show that $\forall (y_n)\subset Y, \:\: y_n \rightarrow0 \:\:\: \exists (x_n) \subset X \:\:x_n\rightarrow0:f(x_n)=y_n, \:\: n=1,2,\dots$ I know that all…
luka5z
  • 6,359
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1 answer

Why is $T:\ell^1\to(\ell^\infty)'$ isometric

The map $T:\ell^1\to (\ell^\infty)', (Tx)(y)=\sum_{n=1}^\infty x_ny_n$ is isometric, but not surjective. According to my book it is easy to prove that $T$ is isometric, but I don't quite know how to show this. I think we have to show…
dinosaur
  • 2,252
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Density of $C^\infty(\overline{\Omega})$ in $L^2(\Omega)$: can we find a bounded sequence approximating $a \in L^2(\Omega)$

Let $a \in L^2(\Omega)$ (bounded $\Omega$) with $0 \leq a(x) \leq C$ a.e. We know $C^\infty(\overline{\Omega})$ is dense in $L^2(\Omega)$, so there exist smooth functions $a_n \to a$ in $L^2$. But can we find a sequence $a_n$ such that $0 \leq…
riem
  • 761
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1 answer

Span of a closed subspace

Let $Y$ be a closed set of a Banach Space $X$. Is it true that the linear Span($Y$)is also closed? For the examples I have tried, I see that the result holds true. I understand that the linear span of any set is dense in the closed linear span of…
G.Dinesh Nathan
  • 125
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1 answer

Eigenvectors of a Multiplication Operator

Let $\mu$ be a positive finite Borel measure on $[-\pi,\pi)$. Define the multiplication operator $M_\mu : L_2(\mu) \to L_2(\mu)$ by $(M_\mu f)(\theta) := e^{i \theta} f(\theta)$. I've proved that $M_\mu$ is an unitary operator; so that each of its…
ragrigg
  • 1,675
2
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2 answers

Doesn't this theorem hold for general normed spaces

My question is: does this hold for any normed space $X$ or only for Banach spaces: If $X$ is a Banach space then $K(X)$ (space of compact operators) equals $B(X)$ (space of bounded operators) if and only if $X$ is finite…
Student
  • 1,687
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1 answer

Operator, closed

We consider the concrete Hilbert space $L^{2}(E,m)=L^{2}(E,\mathcal{B},m)$ with usual inner inner product $(\cdot,\cdot)$ where $(E,\mathcal{B},m)$ is a measure space. For $u,v:E\to \mathbb{R}$,we set \begin{eqnarray*} u\vee v:=\max(u,v),\,u\wedge…
ko4
  • 541
2
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1 answer

quotient map surjective and linear

Let $Q:X\to Y$ be a quotient map, i.e. a map between normed spaces such that $B_Y(0,1)=Q(B_X(0,1))$ ($Q$ maps the unit balls onto each other). Then $Q$ is surjective and $Q\in L(X,Y)$ with $\|Q\|=1$. I already showed that $Q$ is surjective: Let…
dinosaur
  • 2,252
2
votes
1 answer

Prove about weak convergence in $C[0,1]$

I have to prove that (1) if a function sequence $(f_n) \subset C[0,1]$ is weakly convergent to $f\in C[0,1]$ then $f_n(x)\rightarrow f(x)$ for any $x \in [0,1]$. I have also to (2) show, that pointwise convergence of $(f_n) \subset C[0,1]$ does not…
luka5z
  • 6,359
2
votes
1 answer

Properties of a Minkowski sum

Why is it necessary for one of the sets in the Minkowski sum to be bounded (given that both sets are closed) in order that the Minkowski sum be closed? [Edit: In view of @robjohn 's comment (thanks!), I should perhaps change the question to: "Why…
bart
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