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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Embedding $L^2[0,1]$ into any Hilbert space?

Is it true that every Hilbert space has a closed subspace isometrically isomorphic to $L^2[0,1]$? Can someone sketch a proof of this, or at least point me in the right direction to understanding it? ...
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Theorem about equivalent norms.

Let $\|\cdot\|_1$ and $\|\cdot\|_2$ be equivalent norms. Then (i) $\|x\|_1<1$ iff $\|x\|_2<1$; $\|x\|_1>1$ iff $\|x\|_2>1;$ (ii) $\|x\|=1$ iff $\|x\|_2=1$. I want to prove this using ...
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1answer
25 views

Compact Operators: Approximation Property

Note: This thread is not to gain reputation!! Given a Hilbert space. It has the approximation property!
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16 views

Approximation Property: Characterization

Problem Given a Banach space. Suppose it has the approximation property: $$C\in\mathcal{C}:\quad\|T_\varepsilon-1\|_C<\varepsilon\quad(T_\varepsilon\in\mathcal{F}(E))$$ Then every compact ...
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1answer
23 views

about a sequence of isometries' convergency.

Let $M$ be a compact metric space, let $(i_n)$ be a sequence of isometries: $M \rightarrow M$. I've already showed that there exists a subsequence $(i_{n_k})$ that converges to $i$ which is also a ...
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0answers
10 views

When is the completion of a topological vector space a Frechet space?

Suppose $X$ is a topological vector space with the metric topology. If we take the completion of $X$ with respect to the metric, will we get a Frechet space? Are there any extra conditions needed to ...
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20 views

Prove that any function f in $L^p$ is the limit, in the metric of $L^p$, of a sequence of simple functions. [on hold]

I know that I'll need to use dominated convergence here. In the problem, they ask to consider, when f is bounded and nonnegative, the sequence: $s_n(x) = \begin{cases} \frac{i-1}{2^n} \text{ for } ...
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1answer
21 views

If $f_n \rightarrow f$ in $L^p$ and $g_n \rightarrow g$ in $L^q$, where $\frac{1}{p} + \frac{1}{q} = 1$, show that $f_n g_n \rightarrow fg$ in $L^1$ [duplicate]

I know that this will have something to do with Holder's inequality but I am at a loss as to how the $L^p$ and $L^q$ convergence in $f$ and $g$ dictate the convergence in $L^1$. Any help is ...
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1answer
31 views

Inverse operator of $I-A$

Let $H$ be an Hilbert space, $A:H\to H$ be a bounded linear operator such that $$ \|A^{n_0}\|< 1\qquad\text{for some}\quad\; n_0\in\mathbb{N}. $$ I have to show that $I-A$ is invertible. My idea ...
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19 views

Check Functional Analysis Proof

I seem to have proved something with elementary techniques even though the paper I found it in suggests it requires heavy tools. There could be a mistake but I can't find it if there is one. Theorem: ...
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1answer
21 views

weak -star topology on ball M[0,1] [on hold]

let B = ball M[0,1] and for $ \mu$, $v$ in M[O,1] define $ d( \mu, v) = \sum_{n=0} ^{\infty} 2^{-n} \vert \int _0 ^1 x^n d \mu - \int _0 ^1 x^n d v \vert $ .Show that d is a metric on M [0, 1] that ...
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1answer
14 views

Partial Isometries: Subspaces

Note: This thread is not to gain reputation!!! Given an operator algebra. Then a partial isometry satisfies both: ...
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2answers
51 views

Series convergence and compact space

Let $K$ be a compact topological Hausdorff space. $\{x_n\}_1^\infty \subset K $ such that $x_i \not= x_j, i \not=j$ and $\{a_i\}_1^\infty \subset \mathbb{K}$. Show the folowing are equivalent: for ...
2
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1answer
18 views

Adjoints of operators between different Hilbert spaces.

When we have an operator $$ T ~\colon \mathscr{H} \longrightarrow \mathscr{H} $$ from a Hilbert space to itself, we can use the Riesz representation theorem to prove the existence of the adjoint ...
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1answer
12 views

Partial Isometries: Characterization

Note: This thread is not to gain reputation!!! Given a C*-algebra. Any partial isometry satisfies: $$WW^*W=W$$ From this, one derives projections: $$W^*W,WW^*$$ Conversely, given projections: ...
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$C(K)^*$ when K is a countable, compact metric space. [on hold]

If K is a countable, compact metric space, then why $C(K)^*$ consists of only purely atomic measures? Also, why $C(K)^*$ is isometric to $\ell_1$ ? (See Topics in Banach space theory by Albiac and ...
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7 views

Separating and cyclic vector

Let $\{\Gamma_i , \mu_i\}_{i\in I}$ be a family of probability measure spaces and suppose $I$ is uncountable. Let $\{\Gamma , \mu\} = \prod_{i\in I} \{\Gamma_i,\mu_i\}$ be the product measure space. ...
2
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2answers
33 views

Logic behind a proof in Topological Vector Spaces

I found the following result at the beginning of some notes on topological vector spaces (TVS). This is a quite fundamental result, that apparently is considered the corresponding version of the ...
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1answer
16 views

Semi-norm on essentially bounded functions

Consider the space of essentially bounded functions (before quotienting it to create the $L^\infty$ space). On that space, I read, $|||.|||_\infty$ is only a semi-norm. So I wanted to find an example, ...
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1answer
16 views

Direct sum and intersection of sets [on hold]

If $A_1,A_2,B_1,B_2$ are subspaces of a Hilbert space $H$, is the following statement true or not? $$\left( A_1 \oplus A_2\right)\cap \left( B_1 \oplus B_2\right)=\left(A_1\cap B_1\right)\oplus ...
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0answers
17 views

Finding roots of a discrete complex valued function [on hold]

I am struggling with a numerical problem. I have a discrete dataset with complex valued numbers which are the function of a real variable. The function is a black box. Is there any way to find the ...
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0answers
13 views

Irreducible representation and rank one projetion

Let $A$ be a C*-algebra with a nonzero minimal projection $e$. a - Show that if $\{\pi, H\}$ is an irreducible representation of $A$ such that $\pi(e) \neq 0$, then $\pi(e)$ is a projection of rank ...
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1answer
12 views

A symmetric algebra that is not a C* algebra

Recall that a commutative Banach $*$-algebra $A$ is called symmetric if the Gelfand transform replaces involution in $A$ by complex conjugation in $\mathbb{C}$. Moreover, any commutative C* algebra is ...
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Prove that $C_c(X)^c=C_0(X)$ if $X$ is locally compact $T_2$ [on hold]

Prove that $C_c(X)^c=C_0(X)$ if $X$ is locally compact $T_2$ So in genarel $C_c(X)$ is not complete wr.t . Supnorm.
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1answer
26 views

Prove that the space of sequences with limit $0$ is complete.

Prove that $C_0$ (the space of sequences with limit $0$) is complete. My effort: Let {$x_n$} be sequnce in $C_0$ converging to the limit $0$. As the {$0$} is in $C_0$ hence $C_0$ closed in $C$ ...
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2answers
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Is this a bounded linear map?

I tried very hard to (dis)prove it, but now I give up. Define a map which maps $x\in L_2[0,1]$ to the function $$(Tx)(t) = \frac{1}{\sqrt{t}}\int_0^t \frac{x(s)}{\sqrt{s}} \,d s.$$ I don't even ...
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1answer
15 views

How to compute the $H^{-s}(\Omega)$-norm of a function?

Suppose to have a sufficiently regular domain $\Omega\subseteq\mathbb{R}^d$. I know that, for $s\in\mathbb{R}_+$, the space $H^{-s}(\Omega)$ is defined as the dual of $H^s_0(\Omega)$, endowed with the ...
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0answers
25 views

$-\Delta u = f$ in $L^2(0,T;H^{-1}(\Omega))$ (as opposed to $H^{-1}(\Omega)$)

Why does nobody consider the equation $-\Delta u = f$ in the space $L^2(0,T;H^{-1}(\Omega))$? Eg. given $f \in L^2(0,T;L^2(\Omega))$ find a solution $u \in L^2(0,T;H^1_0(\Omega))$ such that ...
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Is this a Hilbert space? If not, is it reflexive?

Let $E$ be a Banach space. Let $L^2(\Omega, E)$ denote the space of random variables taking values in $E$ with second order moment. Is $L^2(\Omega,E)$ a Hilbert space? or at least, reflexive? 1) I do ...
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21 views

Show that elements $u \in W^{1,\infty}(U)$ have continuous representatives

Let $U$ be a bounded, open subset of $\mathbb{R}^n$, and suppose that the boundary $\partial U$ is of class $C^1$. Suppose that $u \in W^{1, \infty}(U)$. I wish to prove that there exists a ...
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19 views

Weak star convergence question

Let $C$ be a convex cone in $L^{\infty}$, that is if $x,y \in C$ and $\alpha, \beta > 0 $ then $\alpha x + \beta y \in C$. Let $U$ be the unit ball in $L^{\infty}$. Assume that for each sequence ...
3
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0answers
23 views

I want to know that the supremum function continuous [on hold]

Let $g(y)=\sup_{x\in[0,y]}f(x)$ for $y\ge0$. I want to know that the function $g(y)$ is continuous on $[0,\infty)$. (here we suppose $f(x)$ is continuous on $[0,\infty)$)
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2answers
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If $X$ is compact then prove that $C(X)=C_0(X)=C_c(X)$

If $X$ is compact then prove that $C(X)=C_0(X)=C_c(X)$. My effort: We know that for any $X$, $C(x)\supseteq C_0(X)\supseteq C_c(X)$. Now to prove the reverse let $f\in C(X)$, Then $f$ is ...
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Analyze the variation of this function $f(x)=x \exp[a+b(x-1)] \, E_1[a+b (x-1)]$ w.r.t. $x$

Please, I need to analyse the variation of the following function w.r.t. $x$ : $f(x)=x \exp[a+b(x-1)] \, E_1[a+b (x-1)]$, where $E_1[a+b (x-1)]$ is the exponential integral, $b>a$, $a>0$, ...
1
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1answer
29 views

Linear Transformations on Infinite Dimensional Vector Spaces

Let $T$ be a linear transformation $T:V\to V$, where $V$ is an infinite dimensional vector space. How can we construct examples such as $1.$ T is one to one but not onto $2.$ T is onto but not ...
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1answer
26 views

Boundedness and Schrödinger operators

I'm working on Schrödinger operators, currently struggling with the proof of $\nabla(\Delta + i)^{-1/2}$ bounded. The domain is $H^2(\mathbb R^3)$. So, of course $(\Delta + i)^{-1}$ is bounded, and ...
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27 views

Weak solution of elliptic equation depends continuously on parameter

Suppose I have a weak formulation of the form: find $u \in H^1_0(\Omega)$ such that $$\int_\Omega b(p)(\nabla u \nabla v + \lambda uv)=0$$ holds for all $v \in H^1_0(\Omega)$ where $b:[a,b] \to ...
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0answers
15 views

Time derivative of a function involving absolute value

I have a functional looking like this $$ L[u] = \int \int_{\Omega} k \left| \nabla u \right|^2$$ , for which I want to take the time derivative $\dot{L}$. I am not sure that I handled the time ...
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1answer
26 views

Pre-dual of distributions with support in a closed subset

Usually, in order to define the space of distributions $\mathcal{D}'(\Omega)$ on an open subset $\Omega \subseteq \mathbb{R}^n$, one considers the space $\mathcal{D}(\Omega)$ of $C^\infty$-test ...
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1answer
27 views

Counterexample Poincaré Inequality for $H_0^1$ in 2D

Is there any counterexample to the Poincaré inequality $$\int_\Omega|f|^2dx\leq C(\Omega)\int_\Omega|\nabla f|^2dx $$ for $f\in H_0^1(\Omega)$, $C(\Omega)>0$ and $\Omega\subset\mathbb{R}^2$? I ...
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1answer
40 views

Gateaux but not Frechet differentiable functional

For functional between Banach spaces X,Y: By Gateaux differentiable at $u\in X$ I mean that there exists bounded linear operator $dF(u)$ s.t. $F(u+t\xi)-F(u)=dF(u)\xi+o(t)$ for all $\xi\in X$. For ...
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1answer
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>>A. $X$ banch. >>B.M banach. >>C. $X/M$ banach.

I want to prove that any two will imply the other: A. $X$ banch. B.M banach. C. $X/M$ banach. I have proved A+B imply C by taking M closed in X. I am not understanding ...
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1answer
23 views

L1 norm less than BV norm

I will appreciate any hint on this Prove that if $f$ is a function of bounded variation then $\|f'\|_{L_1} \leq \|f\|_{BV}$. When $f$ is differentiable just by the Fundamental Theorem of ...
2
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1answer
28 views

Homeomorphisms of different spheres

It is pretty straightforward to show that if $X$ is a Banach space under two equivalent norms, then the respective open unit balls $B_1$ and $B_2$ are homeomorphic only by showing that $B_i$ is ...
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1answer
45 views

If two linear functionals are such that the kernel of one is contained in the kernel of the other, then they are proportional [duplicate]

Let $V$ be a vector space over $K$ and let $f,g \in V^*$ and satisfy $\ker f \subseteq \ker g$. Show there exist such $c \in K$ so that $c \cdot f =g$ How to approach this problem ?
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2answers
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Elliptic differential operator

I am given the differential operator $D(f):=-(fg)'+hf$ and $D^* (f) = g \cdot f' + hf$ where $h,g$ are some smooth functions and want to find out under which conditions, these two operators are ...
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52 views

Product rule in fractional Sobolev spaces

I have to prove the following inequality $$\Vert fg\Vert_{H^s(\mathbb{R}^3)}\leq C \Vert f\Vert_{H^\sigma(\mathbb{R}^3)}\Vert g\Vert_{H^{s+1}(\mathbb{R}^3)}$$ with $s\geq 0$, ...
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1answer
23 views

Is any bounded linear operator of dual spaces is dual of a linear operator?

Let $X,Y$ be two Banach spaces and $S:Y^* \to X^*$ be a bounded linear operator. Is there always bounded linear $T: X\to Y$ such that $S=T^*?$
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23 views

PDEs, infinite vector spaces, and functional analysis

Suppose I have a PDE for some quantity $q_{t}$:$= q(\boldsymbol{x},t)$ at some time $t$ on a 2D vector space $\boldsymbol{x} = (x_{1},x_{2})$, that looks, for the sake of the question, something like ...
2
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2answers
37 views

Is the composition function again in $L^2[a,b]$ [on hold]

Let $f \in L^2[a,b]$. 1- In what condition(s) on a function $g:[a,b]\rightarrow [a,b]$ we can get $$f \circ g \in L^2[a,b]?$$ 2- In what condition(s) on $g:[a,b]\rightarrow [a,b]$, the operator ...