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)

0
votes
1answer
64 views

Does a clean proof exist of why $\mbox {dim range}T=\mbox {dim range}T^*$?

Does a clean proof exist of why $\mbox {dim range}T=\mbox {dim range}T^*$? Here T^* is the adjoint or the conjugate transpose. This is assuming we have a finite dimensional vector space such that $T ...
0
votes
1answer
40 views

What allows us to take the adjoint of both sides of an equation?

Suppose that I have that $ST = TS =I$ Where $T,S \in L(V)$ What allows me to take the adjoint of all three sides to get: $T^*S^*=S^*T^*=I^*=I$
2
votes
1answer
163 views

Dual of Hilbert space dense in dual of Reflexive space.

I don't see how to solve this problem which I think should be easy: Let Y be a reflexive space. Assume $Y$ is continuously embedded in a Hilbert space $H$ and $Y$ is dense in $H$. Show that $H^*$ is ...
1
vote
1answer
53 views

spectral measure and integral query

I have proved the 'resolution of the identity' for a normal operator, namely that there is a unique spectral measure E such that $\int_{{\sigma}(T)} {\lambda}\,dE=T$ If (${\lambda}_{n}$) is the ...
1
vote
1answer
76 views

Show that $\lim_{p \to \infty}||x||_p=||x||_M$ [duplicate]

Let $$||x||_p=\left( \displaystyle \sum_{i=1}^n|x_i|^p\right)^{\frac{1}{p}}$$ and $$||x||_M=\max\{|x_1|,|x_2|,...,|x_n|\},$$ norms in $\mathbb{R}^n$. Show that $$\lim_{p \to \infty}||x||_p=||x||_M, \ ...
1
vote
0answers
85 views

Analytic family of operators?

If $E_{\lambda}, F_{\lambda}$ are two families of complex Hilbert spaces and $L_{\lambda} : E_{\lambda} \rightarrow F_{\lambda}$ is a family of bounded linear operators, where $\lambda$ is a complex ...
1
vote
3answers
308 views

Borel functional calculus

For a normal operator T, we have a resolution of the identity $\int_{{\sigma}(T)} {\lambda}\,dE=T$. If $T$ is in addition compact , we have that $\sum_{n=1}^{{\infty}}{\lambda}_{n}\langle ...
2
votes
0answers
160 views

Equivalence of two norms in complete space

Let $X$ be a vector space with two norms $\| \cdot \|_1$ and $\| \cdot \|_2$ such that $\| x \|_1$ $\leq$ $\| x \|_2$ for all $x \in X$. If $X$ is complete in both norms, prove they are equivalent. ...
2
votes
1answer
149 views

Show that $L^1\subsetneq (L^\infty)^*$ [duplicate]

How does one show that $L^1\subsetneq (L^\infty)^*$? I am having trouble in this. Any help would be appreciated.
6
votes
2answers
185 views

$L^\infty((0,T)\times\Omega)$ is not equal to $L^\infty(0,T;L^\infty(\Omega))$.

Let $\Omega$ be bounded domain in $\mathbb{R}^n$. We know that $L^\infty((0,T)\times\Omega)$ is not equal to $L^\infty(0,T;L^\infty(\Omega))$. Are there any circumstances in which we can say that ...
1
vote
1answer
96 views

The operator matrix on Hilbert space

Let $H$ be a Hilbert space and $P$ be the projection operator, then $H= P(H)\oplus (1-P)(H)$. Hence, for each $T\in B(H)$, we have $$T=\left(\begin{array}{ccc} PTP & PT(1-P) \\ (1-P)TP ...
0
votes
1answer
57 views

If $f \in H^1$ and $f=g$ a.e. or in $L^2$, is $g \in H^1$?

If $f \in H^1(\Omega)$ and either $$f=g \quad\text{a.e.}$$ or $$f=g \quad\text{in $L^2(\Omega)$},$$ is $g \in H^1(\Omega)$? I think since we identify functions that are equal almost everywhere that ...
1
vote
1answer
176 views

Why each nonempty weakly open set of an infinite dimensional normed linear space is unbounded with respect to the norm

Suppose $V$ is an infinite dimensional vector space, $f_i$ ($i$ is from $1$ to $n$) are real-valued linear functions on $V$, I cannot understand why the intersection of kernels of $f_i$ must contain ...
0
votes
2answers
336 views

Core for an unbounded operator.

A symmetric operator $T$ is called essentially self-adjoint if its closure $T$ is self-adjoint. If $T$ is closed, a subset $D \subset D(T)$ is called a core for $T$ if $\overline {T\upharpoonleft D} ...
0
votes
1answer
34 views

$\inf$ of a sequence of $\sup$

let $f_n$ a sequence of right-continuous functions $[0,\infty) \to R$. Assume that for each $T$: \begin{align} & \sup_{\substack{t \in [0,T]\\n \in N}} |f_n(t)|<\infty \end{align} Is it true ...
2
votes
1answer
79 views

convergence of a series in the space of bounded linear operator

I need help in showing that: If $X$ is a Banach space and $T \in L(X,X)$ have $||T||<1$. Use the completeness of $L(X,X)$ to show that $\sum_{n=0}^{\infty}T^n$ converges in $L(X,X)$. where ...
3
votes
1answer
80 views

A easy question on projection operator

Let $H$ be a Hilbert space and $B(H)$ be all the bounded linear operators on $H$, for arbitrary $T\in B(H)$, if $\{P_{i}\}$ is an increasing net of finite-rank projection, can we conclude $P_{i}TP_{i} ...
3
votes
1answer
168 views

Projection operator in Hilbert space

Let $H$ be a Hilbert space, can we find an increasing net of finite rank projections which converge to the identity in the strong operator topology? And I think if $H$ is separable, we can find an ...
1
vote
1answer
72 views

Uncompletion of a Banach Space

Given any Banach space there is a way to define a norm such that is no longer complete. I know you can reach the result by using a Hamel base $H$ for this given space and doing this: If $$ \forall ...
4
votes
1answer
137 views

Generalization of Hahn-Banach Theorem

Let $Y$ be a subspace of a locally convex topological vector space $X$. Suppose $T:Y\longrightarrow l^\infty$ is a continuous linear operator. Prove that $T$ can be extended to a continuous linear ...
1
vote
1answer
36 views

Is a set of jointly bounded functions over a compact domain compact under p-norm?

Let $X$ be a metric space and a measurable space. Let $K$ be a compact set of nonzero measure and $r> 0$. Is a set $\{ f: K\rightarrow \mathbb R| |f|\leq r$ almost everywhere$\}$ compact with ...
1
vote
0answers
66 views

The Haar basis ,proof of orthonoramality.

please i have this problem and i known how to prove completeness but do not know how to prove that it is orthonormal. I will appreciate it if anyone can help me. Given that $n\geq1$ write ...
2
votes
1answer
44 views

Closed ball in $l_p$ is also closed in $l_q$

I am trying to figure out if a closed unit ball in $l_p$ is also closed in $l_q$ for $1 \le p < q < \infty $. It looks easy at a first glance, but I got stuck pretty soon. I supposed there's a ...
1
vote
1answer
60 views

Fredholm operators in Hilbert spaces

Suppose $T_r$ and $T_l$ are the left and the right translations in $l_2$. $T_l$ maps $(x_1,x_2,x_3,...)$ to $(x_2,x_3,x_4,...)$, $T_r$ maps $(x_1,x_2,x_3,...)$ to $(0,x_1,x_2,...)$. It can be easily ...
1
vote
1answer
53 views

Differential equations and surjectivity of some linear operators

Let $a_0,a_1,...a_{n-1}$ be some continuous functions $[0,1]\longrightarrow \mathbb{R}$. Consider a linear operator $D:C^n[0,1]\longrightarrow C[0,1]$ which maps each $y\in C^n[0,1]$ to ...
1
vote
1answer
273 views

Prove that — the range $R(T)$ of a bounded linear operator $T:X\to Y$ need not be closed in $Y$

Prove that the range $R(T)$ of a bounded linear operator $T:X\to Y$ need not be closed in $Y$.
0
votes
1answer
33 views

Calculate the length of the line which represents the function in a plot

Lets say I have a function, that is not linear. I want to calculate the length of the line that I would have drawn if I plot the function. e.g I have the function $f(x)=a x^2$ and I want to ...
1
vote
1answer
60 views

Sobolev Inequality with powers

The following is one of the Sobolev Inequalities as stated in Gilbarg & Trudinger's text on elliptic PDEs: $\displaystyle W_{0}^{1,p}(\Omega)\subset L^{\frac{np}{(n-p)}}(\Omega)$ for $p<n$. ...
0
votes
1answer
109 views

Normal completely positive map in C*-algebra

Let $A$ be a C*-algebra, for a linear map $\phi: A\rightarrow M_{n}(\mathbb{C})$, we define a linear functional $\bar{\phi}$ on $M_{n}(A)$ by ...
0
votes
1answer
57 views

Can a linear operator on a Banach space be both open and closed?

Let $X$ be a Banach space and let $P:X\to X$ be a bounded linear projection. Since $P$ is bounded, $P$ is continuous. Since $P$ is continuous, the Closed Graph Theorem says $\{(x,P(x))\,\mid\,x\in ...
2
votes
0answers
68 views

Continuity of a positive preserving operator between C(X) and C(Y)

I've been struggling with this question in Reed and Simon while I'm prepping for quals. Suppose that $T:C(X)\rightarrow C(Y)$ is a positive operator. Prove that T is continuous and $\Vert ...
1
vote
0answers
42 views

A useful criterion in dispersive PDE.

I would like to prove the following theorem: Consider two Banach spaces $X\hookrightarrow Y$ and $1<p,q\leq\infty$. Let $(f_n)_{n\geq 0}$ be a bounded sequence in $L^q(I,Y)$ and let $f:I\mapsto Y$ ...
4
votes
1answer
162 views

Spectral decomposition of normal operator

Define $T$ from $L_{2}(R)$ into itself by $T(f)(t)=f(t+1)$. Show that $T$ is normal and finds its spectral decomposition. I've shown that $f$ is normal (in fact it's unitary) but how do I find its ...
1
vote
1answer
69 views

Does $u_n \rightharpoonup u$ in $L^2(0,T;H^1)$ and $u_n' \rightharpoonup u'$ in $L^2(0,T;H^{-1})$ give something useful?

If $u_n \rightharpoonup u$ in $L^2(0,T;H^1)$ and $u_n' \rightharpoonup u'$ in $L^2(0,T;H^{-1})$ is there any way to extract a strongly convergent subsequence of $u_n$ in some useful space for PDEs? Or ...
1
vote
1answer
47 views

A compactness argument for small high frequencies

I would like to prove the following statement: Let $N\geq 1$, $1\leq q<\infty$ and let be $E$ a relatively compact subset of $L^q(\mathbb{R}^N)$. Then \begin{equation*} \sup_{u\in ...
2
votes
0answers
29 views

How to interpret $\langle (f(u))', v \rangle = \langle f'(u)u', v \rangle$?

We know that if $u$ has a weak derivative $u'$, and if $f \in C^1$ with $f'$ bounded then $(f(u))'= f'(u)u'$. But how interpret the duality pairing $$\langle (f(u))', v \rangle = \langle f'(u)u', v ...
2
votes
2answers
56 views

Boundedness of a sequence of functions

Let $\Omega$ be a bounded domain and $f_n\in L^2(\Omega)$ be a sequence such that $$\int_\Omega f_nq\operatorname{dx}\leq C<\infty\qquad \text{for all}\quad q\in H^1(\Omega),\ ...
1
vote
0answers
22 views

Showing $\langle ((u-c)^+)', (u-c)^+ \rangle = \langle u', (u-c)^+ \rangle$

Let $u \in L^2(0,T;H^1)$ have weak derivative $u' \in L^2(0,T;H^{-1}).$ Let $c$ be a constant. I want to show that $$\langle ((u-c)^+)', (u-c)^+ \rangle = \langle u', (u-c)^+ \rangle$$ where $(f)^+ = ...
1
vote
1answer
118 views

Links between Minkowski metric, Hamming distance and Levenshtein distance

I know that Hamming distance is a particular case of Minkowski metric (with the specific definition of the subtraction). Also it seems that Hamming distance is a particular case of a Levenshtein ...
2
votes
1answer
172 views

Skorohod convergence (space of right continuous functions with left limit)

If $f_n$ is a sequence of functions of the Skorohod Space $D([0,\infty),E)$, where $E$ is a separable Banach space, such that $f_n \to f$ in the Skorohod topology. Is it possible that there exists a ...
0
votes
1answer
80 views

Show that a certain sequence in $l^\infty(\mathbb N) $ does not converge weakly.

Let $V = l^\infty(\mathbb N) $ (space of bounded sequences with sup norm), I want to show that the sequence $ \{\sum_{ m =n }^\infty e_m\}_{n \ge 1} $ does not converge weakly. Here $ e_i = ( 0,..., ...
2
votes
1answer
112 views

Weak convergence in a subspace

Let $V$ be a normed linear space and $W$ a closed subspace of $V$. Suppose a sequence $\{w_{n}\} \subset W$ and $w \in W$ with $w_{n}$ converges to $w$ weakly in $V$. Why must $w_{n}$ converge weakly ...
1
vote
1answer
45 views

Does $C([0,T];H^{s+1}) \cap C^1([0,T];H^{s})=C^1([0,T];H^{s+1})$?

I have asked this question with other problems, but about this part nobody answers. So I want to ask again, and put some details in it. My question is whether the following equality is correct. If it ...
7
votes
1answer
2k views

L1 convergence gives pointwise convergent subsequence

I have been reading Terry Tao's notes on Real Analysis and there's a part he just says, but does not really explain, so I am wondering if someone here would. The notes are ...
1
vote
1answer
49 views

Sequence does not converge weakly in $\ell^{\infty}(\mathbb{N})$

Let $V = c_{0}(\mathbb{N})$, the space of sequences which converge to 0. Then $V^{\ast} \cong \ell^{1}(\mathbb{N})$ and $V^{\ast\ast} = \ell^{\infty}(\mathbb{N})$. Let $a_{n} = (0, 0, \ldots, 0, 1, 1, ...
1
vote
1answer
70 views

Positive definite kernel vs. positive definite function

What is the difference between positive definite kernels and positive definite functions? As I understand it, a positive definite kernel is a positive definite function if it is translation ...
1
vote
1answer
66 views

The density of $(C[0,1],\|\cdot\|_p)$ in $(L^p[0,1], \|\cdot\|_p)$

Let $1\leq p<\infty$. Consider space $C[0,1]$ equipped with the following norm $$\|f\|=\left (\int_0^1 |f(x)|^pdx\right )^{1/p}.$$ Prove that $(C[0,1],\|\cdot\|)$ is an incomplete space by using ...
1
vote
0answers
46 views

Two different ways to generate the topology of convergence in measure

Consider the measure space $(X, \mathcal{B}, \mu)$ where $\mu(X) < \infty$. Let $L(X)$ denote the space of measurable functions on $X \rightarrow \mathbb{C}$. Then one way to define the topology ...
1
vote
1answer
91 views

formula for the norm of a normal operator

In Rudin's Functional analysis, he does a theorem which shows that for a normal operator $\Vert T\Vert=\sup\left\{|\langle Tx,x\rangle|\colon \Vert x \Vert \leq 1\right\}$. Why can't $\Vert x \Vert ...
2
votes
0answers
89 views

nearest point and closed complement of a subspace in norm spaces

It is well-known that in any Hilbert space $H$, each closed subspace $Y$ admits a closed complement $Y^\perp$. This result also implies that there exists a best approximation point to $Y$ for any ...