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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26 views

Absolute convergence does not implies convergence

Find a space and a series that converges absolutely but it does not converges. It is clear that the space can't complete or Banach.
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13 views

Biorthogonal functionals continuous?

If I have a Schauder basis $(x_n)$ of a Banach space $X$. Such that for every $x = \sum_{i=1}^{\infty} a_i x_i$ for a unique sequence $(a_i) \subset \mathbb{R}$. Is it obvious that the functionals ...
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1answer
24 views

when can you extend a map from a Hilbert basis?

Suppose $H$ and $K$ are Hilbert spaces and $H$ has Hilbert basis $h_i$. What is a necessary and sufficient condition for elements $k_i$ of K so that $h_i \mapsto k_i$ extends to a continuous linear ...
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1answer
42 views

When does open and connected imply path-connected?

It's well known that, in $\mathbb{R}^n$: (1) Open and Connected $\Rightarrow$ Path-connected The proof essentially goes through the fact that (2) Every path-connected component will be open. ...
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1answer
17 views

Unit ball on $B(H)$ and the weak -topology

i have the following problem: i can show that the map $d:B(H)\times B(H)\rightarrow \mathbb{R}$ (with $H$ a separable Hilbert space and $(e_n)_{n\geq1}$) given by: ...
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1answer
24 views

Two questions on Banach-valued spaces of integrable functions

Let $V$ be a Banach space with dual $V'$ and suppose that $V$ is included into the Hilbert space $H$ so that the inclusion is continuous and dense. Then after identification $H = H'%$ we have $V ...
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1answer
13 views

stability of essential spectra

Let $X$ be a Banach space. $A$ and $B$ are linear closed and densely defined operators and $\lambda\in\rho(A)\cap\rho(B)$ such that $(\lambda - A)^{-1}-(\lambda - B)^{-1}$ is a Frehholm perturbation ...
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21 views

How to show that the norms are not equivalent on any infinitely dimensional closed subspace of $C[0,1]$ [on hold]

The norms: $\|x\|_0=\sup|x(t)|$ and $\|x\|_1=\sup(|x(t)|+|x^\prime(t)|)$ where $x^\prime$ is derivative of $x\in C[0,1]$ How to show that the norms are not equivalent on any infinitely dimensional ...
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17 views

Coercivity for functional and complete orthonormal system

Consider with $\rho \in W^{1,2}([0,\pi])$ the following functional $$J(\rho)=\frac{1}{2}\int_{0}^{\pi}{\rho^2\,dx}$$ I know that in the $L^{2}([0,\pi])$ the coercivity condition is satisfied, but i'm ...
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0answers
20 views

If $f_{n} \in L^{\infty}$, $ \int_{0}^{1}f_{n_{k}}(x)g(x)dx \rightarrow \int_{0}^{1}f(x)g(x)dx$ for every $g \in L^1$

Supposet that $\{f_{n}\}_{n=1}^{\infty} \in L^{\infty}$. Is the following statement always true? There is a subsequence $\{n_{k}\}$ and a function $f \in L^{\infty}$ such that $$ ...
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1answer
18 views

How to show $(\alpha_n)\in l_\infty$

Here, we have a sequence $(\alpha_n)$, and let $T:l_1\rightarrow l_2: (x_n)\mapsto (\alpha_nx_n)$ is well-defined. How to show that: $(\alpha_n)\in l_\infty$ Could you please help with this ...
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3answers
33 views

Prove that if $\dim X'<\infty$ then $\dim X<\infty$

I have to prove that $\dim X'<\infty$ then $\dim X<\infty$ where $X$ is a normed vector space and $X'$ is a space of all linear and continuous functionals from $X$. How can I prove this? I ...
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1answer
33 views

solve the functional equation

Let $\phi : R-> C $ (complex numbers) $\phi(0)=1$ $ \phi(-t) = \overline{\phi(t)} $ ( continuous and bounded) solve the functional equation: $Re \phi(t)= \phi(t) \overline{\phi(t)}$ This is all ...
4
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1answer
15 views

Extension of character in Banach algebras

Let $A$ be a Banach algebra. The continuous linear functional $\phi:A\to\Bbb{C}$ is called character if it is non-zero multiplicative function i.e., for every $a,b\in A$ we have ...
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3answers
33 views

Matrix norm in Banach space

How can I calculate the following matrix norm in a Banach Space: $$ A=\begin{pmatrix} 5 & -2 \\ 1 & -1 \\ \end{pmatrix} ?$$ I have tried ...
2
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1answer
26 views

Is this set in $C[0,1]$ a countable union of nowhere dense sets?

$\mathcal{A}=\lbrace x=x_y(t)=\int^{t^2}_0 y(\tau)d\tau : ||y||\le 1\rbrace$ space $X=C[0,1]$ $A_n$ is a nowhere dense if the interior of the closure is empty, ...
2
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1answer
20 views

Let $X=L_2(0,1)$ and $S=\lbrace x\in X: \int_0^{1/2}x^2 dt\le1\rbrace$

Let $X=L_2(0,1)$ and $S=\lbrace x\in X: \int_0^{1/2}x^2 dt\le1\rbrace$. I'm trying to show wheter S can be written as $\cup_{n=1}^\infty S_n$ where $S_n\in X$ and $Int\bar{S_n}=\emptyset$ I tried a ...
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0answers
18 views

Prove that $f(\partial D)$ is closed, known $D$ is a open set, $f$ is a compact vector field [on hold]

Let $D$ be a open set in a Banach space $E$. $f$ is a compact vector field from $\overline D$ to $E$. Prove that $f(\partial D)$ is closed. $f$ is a compact vector field means $f=Id_{| \overline ...
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22 views

How to show: $\exists$ a subspace $L$ of a Hilbert space $H$ such that $H\neq L\oplus L^{\perp}$.

How to show: $\exists$ a subspace $L$ of a Hilbert space $H$ such that $H\neq L\oplus L^{\perp}$. Let consider $H=l_2$ where $l_2=\lbrace x=(x_n)^\infty_1: \sum^\infty_1 |x_n|^2<\infty \rbrace $ ...
2
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1answer
20 views

can set $\mathcal{A}$ be written as union of countable set which are rare sets

$\mathcal{A}$=the set of all fnite sequences in $l_1$ $l_1$: the space of sequences of $x_n$ s.t. : $\sum^\infty_1 |x_n|<\infty$ $A_n$ is rare set if the interior of the closure is ...
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1answer
32 views

Weak $L_1$ norm is different from $L_1$ norm on a probability space

Can any one give an example of a probability space $(X , \mu )$ and functions $f_1, ...,f_n : X \to \mathbb R$ such that $ \| f_i \|_{ L^{ 1, \infty }} : = \sup_{ t > 0} t \lambda_{f_i} (t) \le 1$ ...
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1answer
19 views

weak $L_p$ implies bounded integral on finite measure set

Let $(X, \mu)$ be a measure space which is $\sigma$-finite. $ 1 < p < \infty $. $f : X \to \mathbb C$ is a measurable function. If we know $f$ is in the weak $L_p$ space, i.e. $ ||f||_{L^{(p, ...
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29 views

the dual space of $L^p$

I am reading some preliminary material to develop a good background in order to study PDE and I came across the following fact The dual space of $L^p$ is $L^q$ where $q$ is the Holder's Conjugate of ...
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0answers
18 views

The range of $T$ is closed iff the range of $T^*$ is closed [duplicate]

Let $H$ be a Hilbert space and let $T \in \mathcal{L}(H)$. Define the adjoint operator of $T$, $T^*$, as usual. I want to show that the range of $T$ is closed if and only if the range of $T^*$ is ...
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1answer
33 views

Prove that $c$ is separable

I need to prove that $c$ - space of all convergent sequences - is seperable. I believe that $c$ is a subspace of $\ell^1$. Now, $\ell^1$ is separable, so $c$ is also separable. Edited: So let $R$ ...
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0answers
18 views

Equi integrability and weak convergence of measures

Let $f_n$ be a sequence of functions in $L^1(K, m ; \mathbb{C})$, $K$ metric compact and $m$ a Radon measure on $K$. Assume that $\| f_n \|_1 \leq 1$. From what I understand, there is a subsequence ...
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1answer
23 views

Two spaces which are isometric [on hold]

prove that if X and Y are isometric and X is complete,then Y is also complete.Thanks
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1answer
22 views

Trace norm of Hermitian matrix

Let $A\in L(H)$ some Hermitian matrix, where $H$ is some finite dimensional Hilbertspace. I want to show $$\left\|A\right\|_{tr} = \max_{U\in U(H)}|\text{tr}(UA)| \ \ \ (*)$$ where U is unitary, and ...
1
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1answer
12 views

Is the composition of monotone operators monotone?

Let $H$ be a real Hilbert space with inner product $\langle\cdot, \cdot \rangle: H \times H \rightarrow \mathbb{R}$, and induced norm $\left\| \cdot \right\|: H \rightarrow \mathbb{R}_{\geq 0}$. Let ...
1
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1answer
35 views

Showing an operator is linear

Definition: Let $X$ be a Banach space and $I$ the identity operator on $X$. A family $\{T(t)\}_{t\geq 0}$ of bounded linear operators from $X$ into $X$ is a semigroup of bounded linear operator on ...
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1answer
39 views

L1 convergence and Lp bounded implies Lq convergence

I have tried to solve this problem for almost a week and did not manage to, so I figured to ask it here: Let $(u_n)\to u$ in $L^1(0,1)$ strongly and let $\{u_n\}_{n\in\mathbb{N}}$ be bounded in ...
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1answer
40 views

Show that this linear operator is not continuous.

This is a question of my last exam in Functional Analysis of my graduation: Consider that norm: $|f|=\int_0^1 x^2 f(x)$ where $f\in C^0([0,1])$. Show that the linear operator $f(1-x)\in ...
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1answer
27 views

prove f is bounded linear functional

Let $X=C[-1,1]$ and define $f:X\rightarrow R$ by $f(x)=\int_{- \ 1}^{\ 0} x(t) dt-\int_{ \ 0}^{\ 1} x(t) dt$. Show that $f$ is a bounded linear functional.
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32 views

family of functions/sequences taken over reals instead of naturals

How does convergence for a sequence and a family of functions change when considering $n$ taken from $\mathbb{R}$ instead of the $\mathbb{N}$? for example, consider mollifiers which are defined as ...
1
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1answer
39 views

Prove that $\ell^p$ and $C[0,1]$ are infinite dimensional

I think I should use theorem which states that every linear functional from the finite dimensional normed space into normed space is continuous. But for both of these spaces I can find a linear ...
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0answers
38 views

$C^1[0,1]$ is not complete with respect to sup norm

I think the right sequence is $f_n(t)=|t-\frac{1}{2}|^{(n+1)/n}$ but I can't manage to prove it's Cauchy... (when I look at graph of $f_n$s it seems obvious, but I need it formal way.
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1answer
15 views

Is this subalgebra of a semisimple algebra semisimple?

Let $A$ be a semisimple algebra and $e$ be an idempotent in $A$. Then $eAe=\{eae:a\in A\}$ is a subalgebra of $A$ with $e$ as the identity. We want to prove that $eAe$ is also semisimple. That is, if ...
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0answers
23 views

Calderón-Zygmund $\times$ Schwartz $=$ Calderón-Zygmund

I am in a functional analysis class, and we are being asked to show that if $\eta$ is a Schwartz function and $K$ is a Calderón-Zygmund distribution, then their product is also a Calderón-Zygmund ...
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0answers
12 views

Real Hypersurfaces In Complex Manifolds

I have a problem: ================= I don't understand (2.12) and (2.13) :( How to prove that $$PF=\sum_{\min(k,l) \le 1}F_{kl}+G_{11}\left \langle z,z \right \rangle +\left ( G_{10}+G_{01} ...
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14 views

If $f$ is a bounded linear functional and $ \|f \| \neq 0 $ then $f(h) \neq o(h)$

I am working on an example from D. H. Griffel's Applied functional Analysis (p. 309) book. But I just can't seem to understand the justification for the following example in the book?. Claim: If ...
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1answer
48 views

Are These Orthonormal Functions?

Working through a text (self-learning, not homework), I reached this problem: Prove that the set of functions $\psi_n(x)=a^{-1/2}e^{i\pi nx/a}$ is orthonormal for integer $n$. So that's a fairly ...
3
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2answers
41 views

Determining if the span of a set is dense in L^2(0,1)

I am trying to determine whether or not the following statement is true: If $f \in L^2(0,1)$ and $\int_0^1 x^nf(x) = 0$ for all positive integers $n$. Then $f(x) = 0$ I have already verified this ...
4
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0answers
43 views

Extension and trace operators for Sobolev spaces

Given that $\Omega \subset \mathbb{R}^{n}$ is an open, convex, Lipschitz bounded set. Let $O \subset \Omega$ be open bounded set then consider. $$u_{m} \rightharpoonup^{*} u \text{ in } ...
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2answers
28 views

The orthogonal projection over the subspace $V=\{f:f =0\text{ on }[0,1/2]\}$

The orthogonal projection of an element $x_0 \in H$ over a convex set $C$ is the element $y_0 \in C$ such that $\|x_0-y_0\|=\min_{y \in C}\|x_0-y\|$. Find the orthogonal projection of $\gamma ...
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0answers
43 views

Prove that $C^1[0,1]$ is complete

I have to prove that $C^1[0,1]$ is complete with respect to the norm $\|f\|=|f(0)|+\int_0^1|f'(t)|dt$ is complete. My attempt: Let $f_n(t)$ be a Cauchy sequence from $C^1[0,1]$. We have: ...
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1answer
20 views

norm on a quotient-space

Let $M:[0,\infty)\to[0,\infty)$ be continuous and convex. Further $M$ satisfies $M(t)=0\Leftrightarrow t=0$. Let $$\mathcal L_M(\mathbb R):=\left\{f:\mathbb R\to\mathbb R \mathrm{\ measurable\ ...
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0answers
28 views

How to prove that this sequence is Cauchy

How can I prove that $f_n(x)=\left|x-\frac12\right|^{(n+1)/n}$ is a Cauchy sequence in $C^1[0,1]$ with sup norm? It seems obvious when we look at graphs but I need this formal way.I need this for my ...
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0answers
33 views

$K(X,Y)$ is closed in $B(X,Y)$

I tried to prove that the set of compact operators $K(X,Y)$ is a closed subset of the set of bounded operators $B(X,Y)$ where $X,Y$ are Banach spaces. Please can you tell me if my proof is correct? ...
4
votes
1answer
30 views

Sequence spaces

Suppose the sequence spaces $$d \colon=\left\lbrace \left\lbrace x_n\right\rbrace_{n \in \mathbb{N}} \in \mathbb{K}^{\mathbb{N}} \colon x_n=0 \ \text{for almost all} \ n \right\rbrace$$ and ...
0
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0answers
4 views

Rearranging 2 discriminant function to solve for 1 parameter (to derive a decision boundary)

I have a task where I want to classify patterns from 2 classes where the samples are drawn from a bivariate Gaussian distribution. I use the 2 discriminant functions ($g_1$ and $g_2$) to classify the ...