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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Square root of the operator $T$

Find the positive square root of the operator $T$ on $L^2 ([a,b])$ defined by $(Tf)(t) =g(t)f(t)$, where $g$ is a positive continuous function on $[a, b]$.
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125 views

Is $C(\mathbb R)$ Separable?

I'm working on an exercise from Carother's chapter11 of Real Analysis that talking about Space of Continuous Functions: Here, $C(\Bbb R)$ is the set of continuous real-valued functions on $\Bbb R$, ...
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35 views

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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39 views

Knowing a function while only knowing its partial derivatives?

So again we study a physics course without studying mathematics course We are in the work energy chapter , and I'd like to know if you can know the function $f(x,y,z)$ if you know all of its partial ...
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70 views

Parseval's Identity holds for all $x\in H$ implies $H$ is a Schauder basis

Prove that any set $\{v_j\}_{j \in \mathbb{Z}}$ for which the Parseval identity $\|x\|^2=\sum_{j=1}^\infty |\langle v_j,x\rangle|^2$ holds for every $x \in H$ is a Schauder basis. I know that a ...
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54 views

Show that $f \in L^2(\mathbb R)$

Let $1\le p < 2 < q \le \infty$. Show that if $f\in L^p(\mathbb R)\cap L^q(\mathbb R)$, then $f\in L^2(\mathbb R)$.The hint is to use Holder and $a=a-b+b$. I tried to start of with: $\int_R ...
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38 views

Prove $d(x,y)=\sup _{n} \left| \sum _{k=1}^{n}(x_k-y_k)\right |$ is a metric

Let $\gamma$ be the set of convergent series.$$\gamma = \{x=(x_k), x_k \in \mathbb{R} : \sum x_k <\infty\}$$ Prove that $(\gamma , d)$ is a metric space, with $$d(x,y)=\sup _{n} \left| \sum ...
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62 views

Obtain solution of boundary problem as linear operator.

I'm kinda stuck with a problem right now. I have the boundary problem $$\left\{ \begin{array}{l} -u''(x)+\mu u(x)= f(x), \quad x\in (0,T) \\ u'(0)=u'(T)=0 \end{array} \right.$$ and I have to obtain ...
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36 views

Finding multiple functions with same $f_{even}$ but different $f_{odd}$?

A function can be decomposed as $f(x) = f_{even}(x) + f_{odd}(x)$ where $f_{even}(x)=\dfrac{f(x)+f(-x)}{2}$ and $f_{odd}(x)=\dfrac{f(x)-f(-x)}{2}$. If we know only $f_{even}$, how can we find ...
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44 views

A question about inclusion of $L^r(\mu)$ spaces for different $r$ and different measures $\mu$

For some measures, the relation $r<s$ implies $L^r(\mu)\subseteq L^s(\mu)$ ; for others, the inclusion is reversed; and there are some for which $L^r(\mu)$ does not contain $L^s(\mu)$ if $r\ne ...
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86 views

The space $C^1[a,b]$ is complete.

Denote $C^1[a,b]$ to be the set of all $C^1$ functions $f: [a,b] \to \Bbb R$. Prove $C^1[a,b]$ is complete under $\| f \| = \max |f| + \max |f'|$. So my idea is this (1) We know $\| f_n - f_m \| ...
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33 views

A question about spectral measure

The following is a part of a theorem of Takesaki's Operator theory: Let $T$ be an positive operator. Suppose $T = \int_0^{\|T\|} \lambda \, de(\lambda)$ is the spectral measure of $T$. Also put ...
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243 views

Is the set of natural numbers with this metric complete?

Let $\mathbb{N}$, the set of all natural numbers, be given the metric $d$ defined as follows: $$ d(m,n) \colon= | m^{-1} - n^{-1} |$$ for all $m$, $m$ in $\mathbb{N}$. Then how to determine if ...
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172 views

Bounded measurable functions

Suppose $X$ is a compact space and $B(X)$ denotes the bounded Borel measurable function space. Let $f\in B(X)$. There is a sequence of step functions $\{\phi_n\}$ such that $\phi_n\to f$ (point wise). ...
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45 views

Is this a bijective operator?

Let $T: C \left[-1, 1 \right] \rightarrow C \left[-1, 1 \right]$ be the operator defined by $(T \psi) (t) = \int_{-1} ^{t} \psi(s) ds.$ I know it is continous, but dont't know if it is bijective. I ...
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64 views

Find a space whose dual does not separate points

I read about the fact that for a locally convex topological vector space $X$, its dual $X^*$ separates points, i.e. for any $x\neq y$ in $X$, $\exists f \in X^*$ such that $f(x)\neq f(y)$. Could you ...
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40 views

Is there a Lipschitz continuous bijection between a ray and the whole real line?

my question is already in the title, I state it here in an equivalent way: Question: Is there a bijective Lipschitz continuous function $f \colon (0,\infty) \to \mathbb{R}$? I think the answer is ...
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80 views

The limit of $\int_{0}^{1}\frac{\sqrt{n}}{1+n\ln(1+x^2)}dx$ as $n\to\infty$

The task is to calculate $$\lim_{n\to\infty}\int_{0}^{1}\frac{\sqrt{n}}{1+n\ln(1+x^2)}dx$$ I tried various estimates I know to find the dominating integrable function and nothing worked. Does anyone ...
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45 views

Does it make sense to talk about the integral of measurable functions that are not absolutely integrable?

Suppose $f$ is a real-valued (possibly infinite-valued) function on some measure space $(X, \Sigma, \mu)$, and suppose that it is measurable. Note that $f$ is not necessarily nonnegative. Does it ...
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80 views

Reflexivity of $C[a,b]$

I find the statement that the normed, complex or real, linear space $C[a,b]$ is reflexive, i.e. the natural map of the space $C[a,b]$ into the bidual space $C[a,b]^{\ast\ast}$, defined by ...
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114 views

a custom designed cutoff function whose derivative is bounded above.

I am trying to find a $C^\infty$ function $\phi(t)$ with the following properties. $\phi(t) =1$ for $\lvert t \rvert \le 1$ $\phi(t)$=0 for $t \geq 2$ $\lvert \phi'(t) \rvert \le 2 $ I have tried ...
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238 views

Eigenvalues of symmetric elliptic operators

As stated in one of my previous questions already, I have not had so much exposure to theoretical linear algebra. This time, I'm reading a theorem and proof from PDE Evans, 2nd edition, pages 335-356. ...
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49 views

Norm of the multiplication operator $f\mapsto (x\mapsto xf(x))$ on $L^2[a,b]$ [duplicate]

We have a linear operator $T : L^2[a,b] \rightarrow L^2[a,b]$ (with $|a| \le |b|$), $f \mapsto (x \mapsto xf(x))$ Now I shall determine what $\Vert T\Vert$ is. We clearly have $\Vert x \mapsto ...
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13 views

Is $\text{Id} = \chi_{\{ |u| \leq k\}} + \chi_{\{|u| > k\}}$ well defined for $u \in L^p(0,T;L^q)$?

Is the decomposition $$\text{Id}(z) = \chi_{\{ |u| \leq k\}}(z) + \chi_{\{|u| > k\}}(z)\tag{1}$$ well defined for $u \in L^p(0,T;L^q(\Omega))$? I guess (1) holds a.e. So the problem is, is the set ...
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33 views

How to choose, $\phi \in C^{\infty}_{c}(\mathbb R)$ such that its Fourier transform $\hat{\phi}$ is 1 in some neighbourhood of the given point?

Put $C_{c}^{\infty}(\mathbb R)=$ The space of $C^{\infty}$ functions on $\mathbb R$ whose support is compact. Fix $x_{0}\in \mathbb R.$ My Question is : Can we expect to choose, $\phi \in ...
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25 views

Is this condition sufficient to ensure the locally convexity of a function at a given point?

Given $\bar x\in \mathbb R^n$. Let $f:\; \mathbb R^n\to \mathbb R$ be a nonconvex continuous function on $\mathbb R^n$ satisfying the followings (i) $f$ is not differentiable at $\bar x$, (ii) There ...
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53 views

Elegant way to solve this extreme value problem

I want to show that $$ \sup_{(x,y)\in \mathbb{R}^2 \setminus \lbrace (0,0) \rbrace} \frac{(ax+by)^2}{x^2+y^2} =a^2+b^2 $$ where $a,b \in \mathbb{R}$ are fixed (this problem appears when one tries to ...
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78 views

Book on periodic Schrödinger operators

I am looking for good books about the spectral theory of periodic (1-dimensional) Schrödinger operators on a compact interval. A good reference I found was Reed/Simon Analysis of Operators (and a ...
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42 views

Proving $||f+g||\cdot||f-g|| \le ||f||^2+||g||^2$ in a Hilbert Space

Let $f$ and $g$ be vectors in a Hilbert space $H$. Show that $$||f+g||\cdot||f-g|| \le ||f||^2+||g||^2$$ My question is, do i have to rewrite $||f+g||$ as $\sqrt{\langle f+g,f+g\rangle}$ and same ...
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29 views

Boundedness in $(M_{\phi}f)(x)=\phi(x)f(x)$ and $(Kf)(x)=\int_a^bk(x,t)f(t) dt$

I'm currently practicing (for a comprehensive exit exam) how to prove something is bounded. Here are 2 questions I'm concerned about: Let $\phi$ be a continuous function on the interval $[a,b]$. ...
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93 views

Fourier series with respect to orthonormal sequence

Let $H$ be the space of piecewise continuous $2 \pi$-periodic functions on the real line. For $f$ and $g$ in $H$, consider the inner product $<f,g>=\frac{1}{2\pi}\int_{- \pi}^{\pi}f(x)\overline ...
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54 views

$(X,|.|_A)$ is Banach implies $A$ is closed

Let $(X,|.|)$ be a Banach space. We know that if $A:X\to X$ is a closed operator then $(X,|.|_A)$ is a Banach space, where $|.|_A$ is the norm defined by $$|x|_A=|x|+|Ax|$$ Then using the "continuity ...
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160 views

Prove that a closed unit ball in $C[0,1]$ is not weak-compact

I have to prove that a closed unit ball in $C[0,1]$ is not weak-compact. The hint is that I should consider sets: $$V_t=\{f\in C[0,1]:|f(t)|>1/3\}$$ and $$U_t=\{f\in C[0,1]:|f(t)|<2/3\}$$ Now I ...
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21 views

Is this function well defined for any $g\in L^q(\Omega)$?

If $p,q\in(1,\infty)$ such that $\frac 1q+\frac1p=1$, given $g\in L^q(\Omega)$ we difine: $$\Phi(g):L^p(\Omega)\to\Bbb R \\ \Phi(g)(f):= \int_{\Omega}fg$$ I know this is a basic question, but how do ...
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128 views

i need help to prove this problem(functional analysis)

show that the annihilator of a set M in an inner product space X is a closed subspace of X.
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64 views

Linear Projections: Bounded/Continuous?

Are linear (nonorthogonal) projections on (pre) Hilbert spaces necessarily bounded/continuous? (can you give a proof or counterexample)
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About locally convex space

Is a Banach space a locally convex space? Why? Recall A locally convex space is a linear topological space in which the topology has a base consisting of convex sets.
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Example- $l_p$ norm space

$$||x||= {[{\sum _{i=1}^\infty |x_i|^p}]}^{1/p}$$ Is a norm on $l_p$ space :- space of all sequences made of scalars from $\mathfrak C$(filed of complex numbers). To prove that above is norm on $l_p$ ...
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84 views

uniqueness of positive operator

Let $A,B$ be commuting positive operators on a Hilbert space such that $\langle(A-B)(A+B)x,x\rangle=0$ for all $x$ in the Hilbert space. Prove that $A=B$. My attempt: The above implies that $A=B$ on ...
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98 views

Prove $C^1[a,b]$ is a Banach space.

Let $C^1[a,b]$ be the space of continuous differentiable functions on $[a,b]$ equipped with the following norm $$\|x\|=|x(a)| + \sup_{t\in [a,b]}|x'(t)|.$$ Prove that $(C^1[a, b],\|\cdot\|)$ ...
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112 views

A simple question about positive element in C*-algebra

I am reading a book about C*-algebra. There is a quotation below. An $operator~system$ $E$ is a closed self-adjoint subspace of a unital C*-algebra $A$ such that $1_{A}\in E$. The $n \times n$ ...
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181 views

Why do we call Functional Analysis like this?

Functional analysis is 'a kind of mathematical analysis' where the object of study are functions. The tool for studying functions are the operators. A specific type of operators are the functionals. ...
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376 views

Linear surjective isometry then unitary

Basically what I'm trying to show is $\forall h_1, \ h_2 \in \mathscr{H}$ and $U: \mathscr{H} \rightarrow \mathscr{K}$ then $\langle Uh_1, \ Uh_2\rangle_\mathscr{K} = \langle h_1, \ h_2 ...
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177 views

Weak convergence of norms of sequence

Let $x_n \to x$ weakly. My question is: does it hold that $\|x_n\|\to \|x\|$? I haven't been able to work out the answer and I'd appreciate help with it but here are my thoughts: Given the inverse ...
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52 views

Definition of $L^p$ as a set

Looking at Functional Analysis, I know that the space of convergent sequences $\ell^p$ is defined to be $$\ell^p := \left\{\left(x_k\right)_{k=1}^\infty\mid\sum_{k=1}^\infty ...
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56 views

convergence and boundedness

If $f_{n}\to f$ in any norm $\lVert\,\cdot\,\rVert$, then since $\lVert f_{n}\rVert = \lVert f_{n}-f+f \rVert \leqslant \lVert f_n - f\rVert + \lVert f\rVert$, we have that $\lVert f_{n}\rVert$ is ...
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35 views

An exercise about compact operater

If $A \in \mathfrak{B}(H)$ and $H$ is a Hilbert space, $AT=TA$ for every compact operater $T$, show that $A$ is a multiple of the identity operater. I don't what is "multiple of the identity ...
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55 views

A problem about projective operater

Let $P$ and $Q$ be projective on a Hilbert space $H$. Show that $P+Q$ is projective if and only if $\mbox{ran }P \perp \mbox{ran }Q$. The sufficiency is easy. About the necessity, suppose $P+Q$ is ...
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484 views

Polar decomposition normal operator

Let $T \in \mathcal{B}(\mathcal{H})$ be normal. I have to show that there exists a unitary operator $U \in \mathcal{B}(\mathcal{H})$ such that $T^*=UT$ and give necessary and sufficient conditions on ...
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39 views

A property of the canonical inclusion $i: L^2(0,1)\to L^1(0,1)$

Prove that the image of the canonical inclusion $i: L^2(0,1)\to L^1(0,1)$ is a countable union of closed sets with empty internal part. Can anyone give me any idea on the solution? Thank you in ...