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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0answers
18 views
Form of weakly continuous linear functional
This was originally a problem in Stratila and Zsido's "Lectures on von Neumann algebras" (E.1.2). I've spent so much time working on it, and right now I cannot see how the result can be so simple.
...
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4answers
74 views
Proof that $\overline{A \cap B} \subseteq \overline{A} \cap \overline{B}$
Given a topological space ($X$, $\mathcal{T}$) and the following defintions
$\partial A$ := { $x$ | every neighbourhood of $x$ contains points from $A$ and from $X \setminus A$ }
$\overline{A}$ := ...
1
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1answer
77 views
$ \int_{-\infty}^\infty |f| < \infty$. Then $ \lim_{x \to \infty} f(x) =0\;?$ [closed]
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
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2answers
50 views
show that a functional operator $F : X \to \mathbb{R}$ is continous
i have the following homework:
Let $g : \mathbb{R} \to \mathbb{R}$ be continous and $X = C([a,b])$ with the metric
$$
d(x,y) := \sup_{t \in [a,b]} |x(t) - y(t)|
$$
show that $F: X \to \mathbb{R}$ ...
1
vote
1answer
15 views
Differentiation of maps
Let $f: \mathbb{R}^3 \to \mathbb{R}$ and $g: \mathbb{R} \to \mathbb{R}$ be differentiable. Let $F: \mathbb{R}^2 \to \mathbb{R}$ be defined by the equation
$$F(x,y)=f(x,y,g(x,y)).$$
(a)Find $DF$ in ...
5
votes
1answer
68 views
Is $\ell^1 \subset \ell^2$ meagre? [closed]
Possible Duplicate:
Prove $\ell_1$ is first category in $\ell_2$
Consider $\ell^2$ with the topology induced by the usual norm. We can easily prove that $\ell^1 \subset \ell^2$. I am ...
0
votes
0answers
21 views
Properties of $i\nabla+\mathbf{f}$
Let $\mathbf{f}\in L^2_\text{loc}(\mathbf{R}^3)^3$ be a real vector field over $\mathbb{R}^3$, and let $\mathbf{F} = i\nabla + \mathbf{f}$. The components $F_i$ are densely defined symmetric operators ...
0
votes
0answers
41 views
Does $f \in C^{\infty} (\Bbb R^n) $, $g \in W^{1,2} ( \Bbb R^n) $, implies $ f \cdot g = f(g) \in L^2 ( \Bbb R^n) $?
Let $f \in C^{\infty} (\Bbb R^n) $ and $g \in W^{1,2} ( \Bbb R^n) $ . Is it true that
$$
f \circ g \in L^2 ( \Bbb R^n)
$$
2
votes
0answers
27 views
Extension of Choi's theorem on extreme completely positive maps
In this paper Man-Duen Choi gave a criteria for a completely positive map to be extreme. For convenience I am writing it below.
Let $\phi:\mathcal{M}_n\rightarrow\mathcal{M}_m$. Then $\phi$ is
...
3
votes
1answer
45 views
An inequality in Evans' PDE
In Section $9.2$ Theorem $5$ of Lawrence Evans' Partial Differential Equations, First Edition the author proves that for a large enough $\lambda$, the equation
$$\begin{array}-\Delta u+b(\nabla ...
3
votes
1answer
29 views
$f \in W^{m,2} ( \Bbb R^n) \cap C^{\infty} (\Bbb R^n )$ then $\| f \|_{L^\infty} < \infty $?
If $f \in W^{m,2} ( \Bbb R^n) \cap C^{\infty} (\Bbb R^n ) $ for $m = 0,1,\cdots$, then can I conclude that $$ \| f \|_{L^\infty} < \infty \;?$$
4
votes
1answer
56 views
Showing closure in the $\|\cdot\|_1$ norm
Let $X=C[0,1]$ and $W=\{f\in X\mid f(0)=0\}$. What is the closure of $W$ wrt the 1-norm $\|\cdot\|_1$.
My solution is as follows:
The closure is the whole space $X$. To see this take any ...
0
votes
1answer
24 views
How can we define multi-dimensional norms?
How can we define multi-dimensional norms?
For example,
$$ \| (v_1, v_2, \cdots , v_n) \|_{W^{1,2}(X)} \;\;\text{or} \;\;\|(v_1 , v_2 , \cdots , v_n ) \|_{L^2 (X)}$$ for some appropriate functions ...
1
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0answers
18 views
A deterministic corollary from a probailistic result for IFS
Let $(X,\rho)$ be a separable complete metric space. We consider an Itereted Function System with place-dependent probabilities. Namely, assume that $I=\{1,\ldots,N\}$, $S_i: X\rightarrow X$ and $p_i: ...
1
vote
1answer
74 views
Iterated duals of Banach spaces
Let $B$ be a (non-reflexive) Banach space. Denote by $B^{(n)}$ the $n$-fold dual, i.e. $B^{(0)}=B$ and $B^{(n)}=(B^{(n-1)})^\prime$. Does there exist an $n\ge 1$ s.t.
$B^{(n)}\cong B$ ? Please ...
