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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12 views

Banach algebra norms on $M_n(A)$

Let $A$ be a Banach algebra (not sure whether I need $A$ to be unital). I saw the claim that all Banach algebra norms on $M_n(A)$ with continuous projections on entries are equivalent. How does one ...
1
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0answers
7 views

Tensor products of Lipschitz functions

I have encountered a problem on which I am sure there is some background, which unfortunately I don't know anything about (so that I don't even know where to start). Let $(M, d_M)$, $(N, d_N)$ be ...
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1answer
9 views

Showing that a bilinear form is coercive

Let $I = (0, 1)$ and $H_0^2 (I)$ the closure of $C_c^\infty (I)$ in $W^{2, 2} (I)$. Consider $$a : H_0^2 (I) \times H_0^2 (I) \to \mathbb{R}$$ defined by $$a(u, v) = \underset{I}{\int} u''(x) v''(x) ...
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1answer
21 views

Showing a set is dense in Hilbert space $ L^2 [0, 2\pi]$. [on hold]

Why the set $ \{ f \in C [0, 2\pi]: f(0) = f(2\pi) \} $ is dense in Hilbert Space $ L^2 [0, 2\pi]$?
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1answer
11 views

Invertible operators converging to a noninvertible operator in a finite dimensions: Eigenvalue converge to 0?

I feel like this should be an obvious property, but I want to make sure of it before I use it as the key part of a larger proof: If we have two finite dimensional vector spaces $E,F$ of the same ...
2
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2answers
23 views

A function that integrates to zero against a sequence of weights

Fix any $a\in(0,1)$. Is there a nontrivial continuous function $f:[a,1]\to\mathbb R$ so that $$ \int_a^1t^{-2n}f(t)dt=0 $$ for all integers $n\geq0$ and $f(a)=f(1)=0$? I would prefer explicit ...
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0answers
16 views

Criterion for Isometry

Let $X$ be a topological vector space, with $d$ an invariant metric compatible with the metric. Let $f:X\to X$ be an involutive linear isomorphism. How do you show that $f$ is an isometry? I ...
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0answers
33 views

definition of largest circle containing in a convex body

For a convex body B(compact convex set with non-empty interior) what does each of these following values mean? 1- $$\ \sup_{x\in B}\inf_{y\in cB} d(x,y) $$ and 2- $$\ \sup_{x\in B}\inf_{y\in ...
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3answers
174 views

Why are we defining the norms on certain vector spaces the way they are?

What's the intuition behind defining $\|x\|_{\infty} = \max_{1 \le i \le n}\{|x_i|\}$ on the space of ordered $n$-tuples of complex numbers? I'm asking because I've been asked to find a norm on the ...
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0answers
16 views

Problem on The Multiplication Operator

Let $f$ be a bounded measurable function on $X$ and $M_f$ be the multiplication operator on $L^2$($\mu$).Then prove that $\int$ $fdP=M_f$ where $P$ is the $spectral$ $measure$. I have been trying for ...
2
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1answer
24 views

Prove a sequentially compact metric space is bounded.

Prove that if the metric space $(X, d)$ is sequentially compact, that there exists points $x_0$ and $y_0$ belonging to $X$ such that; $$d(x, y) \leq d(x_0, y_0)$$ for every $x$ and $y$ belonging to ...
0
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2answers
53 views

Prove $\ell^5$ is contained in $\ell^6$.

I am struggling with the proof to show that, for any $p$, $r$ such that $1 \le p <r < \infty$, that $\ell^ p\subset\ell ^r$. Could somebody please give a helpful nudge by showing how this ...
-2
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1answer
31 views

Prob. 3, Sec. 3.3 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS

Can we find an example where $\mathbb{R}^3$ is a direct sum of two subspaces that are not orthogonal? A vector space $X$ is said to be a direct sum of two of subspaces $Y$ and $Z$ of $X$ if each $x ...
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1answer
34 views

Prob. 2, Sec. 3.3 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS

Let $M$ be the subset of $\mathbb{C}^n$ such that $M$ consists of all $n$-tuples of $y = (\eta_1, \ldots, \eta_n)$ of complex numbers such that $\sum_{i=1}^n \eta_i = 1$. Then we can show that $M$ is ...
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1answer
24 views

Prob. 1, Sec. 3.3 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS

Let $H$ be a Hilbert space, $M \subset X$ a convex subset, and $(x_n)$ a sequence in $M$ such that $\Vert x_n \Vert \to d$, where $d = \inf_{x \in M} \Vert x \Vert$. How to show that $(x_n)$ converges ...
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0answers
15 views

Calculating the norm of a case dependent function

If $X$ is the Hilbert space $L^2(0,\infty)$ equipped with the inner product $$\langle f,g\rangle :=\int_0^\infty f(\zeta)\overline{g(\zeta)}(e^{-\zeta}+1) \, d\zeta,$$ and the operator $T(t):X\to ...
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0answers
7 views

Generating a contraction semigroup on an energy space

Consider the system of partial differential equations $\displaystyle\frac{\partial Q}{\partial t}(\zeta, t)=-\frac{\partial}{\partial\zeta}\frac{\phi(\zeta, t)}{L(\zeta)}$ ...
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0answers
16 views

Fourier transform and series

Let $f \in L^2(\mathbb{R})$ and $F(f|_{[m,m+1]})$ be the Fourier transform of a restriction of $f$. Does this imply that $$\sum_{m,n \in \mathbb{Z}} |F(f|_{[m,m+1]})(2 \pi n)|^2 $$ exists and is ...
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0answers
16 views

Schrodinger Operator with Finite Discrete Spectrum in $(-\infty, -1]$

I'm reading parts of Reed and Simon's Analysis of Operators and have come across a statement I find puzzling. They say that if $V$ is a bounded function of compact support on $\mathbb{R}^3$ then ...
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1answer
7 views

Supporting hyperplane of convex function

Below is the appendix B of Evan's PDE book on supporting hyperplanes of convex functions. In the remark (1), he says that the mapping $y\to f(x)+r\cdot(y-x)$ determines the supporting hyperplane to ...
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0answers
15 views

An exersice about Isomorphic Hilbert Spaces and the Fourier Transform for the Circle [on hold]

An exersice from section 5 of conway's functional analysis
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13 views

When do closed subspaces of a Banach space fit together nicely?

Let $E$ be a Banach space, and let $F_1, F_2, F_3, ...$ be a sequence of closed subspaces of $E$ with $F_i\cap F_j=0$ whenever $i\neq j$. Denote by $\sum_n F_n$ the (not necessarily closed) subspace ...
2
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1answer
25 views

closed graph theory and unbounded operator

I am studying unbounded operators and the graphs of those operators. I found that the closure of a graph may not be the graph of any operator. Can someone provide an example of an operator and a ...
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1answer
33 views

The proof of finding extreme points of the unit ball of $l^1$

Can someone show how to start the proof of finding extreme points of the unit ball of $l^1$? Thanks. Edit: How I've done so far is that Let $B$ be the closed unit ball of $l^1$ Consider any $x_n ...
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1answer
11 views

properties of orthonormal systems and hilbert spaces [on hold]

I need to show (a) $\implies$ (b) For an orthonormal system $\{\phi_i\}_{i=1}^\infty$, and a Hilbert space $H$, the following are equivalent: (a) If $\langle f,\phi_i\rangle=0$ $\forall i$, ...
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votes
1answer
18 views

Polarization Identity: Sesquilinearity

Problem Given a vector space $V$. Consider quadratic forms with: $$q[u+v]+q[u-v]=2q[u]+2q[v]$$ Then one has a 1-1-correspondence: $$q_s[v]:=s(v,v)\quad ...
2
votes
2answers
63 views

What is the largest function whose integral still converges?

Let C be the set of all functions $f(x)$ whose integral converges, i.e. for some constant $x_0$: $$\int_{x_0}^\infty f(x) dx < \infty$$ While playing with integrals in Wolfram Alpha, I noticed ...
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0answers
14 views

fractional powers in Banach algebra [on hold]

Let $X$ be a Banach algebra. For $x\in X$ and $0< p< 1$, would $x^p\in X$? If not, under what conditions it holds?
2
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0answers
20 views

Extreme points of the unit balls of $l^\infty, C([0,1])$

Determine the extreme points of the unit balls of $l^\infty$, and $C([0,1])$ for real-valued functions, with the uniform norm. Is $C([0,1])$ the dual of a Banach space? I've found the extreme points ...
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1answer
18 views

Diffeomorphism preserves compact support of functions?

Let $M$ and $N$ be two Riemannian manifolds which are diffeomorphic via a $C^k$ map $F:M \to N$. Let $\phi \in C^0_c(M)$ be a continuous function with compact support in $M$. Is it true that its ...
5
votes
3answers
83 views

The function $\phi(p)=\|f\|_{L^p}^p$ is convex

Fix an arbitrary function $f\in L^p([0,1])$ and define $$\phi(p)=\|f\|_{L^p}^p$$ for $p\in [1,\infty)$. Prove $\phi$ is convex. Comments: This is a standard property of $L^p$ spaces, but no ...
2
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2answers
57 views

The $L^p(\mathbb R)$ norm is increasing as a function of $p$ (Update: It's false!)

Update: This is false. See the answers for a counterexample. Let $C\ge 1$ be a constant. Fix $f\in L^p(\mathbb R)$ for $p\ge C$. Show that $$p\rightarrow \left( \int |f|^p \right)^{1/p}$$ is ...
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0answers
33 views

Under what condition does $f$ belong to $L^p\left(\mathbb{R}^n\right)$? [on hold]

Let $a,\,b>0$ and $f(x) =(1+|x|^a)^{-1} + (1+|\log|x||^b)^{-1}$ Under what condition does $f$ belong to $L^p\left(\mathbb{R}^n\right)$?
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0answers
40 views

Common traits of functions which are non-trivial to integrate?

My question is very simple: do there exist certain qualities of functions such that functions which possess these qualities are guaranteed not to have anti-derivatives which are expressable in terms ...
4
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0answers
40 views

Is $C([0,1])$ for $\mathbb{C}$ dual to any Banach Space?

I've been able to show that the extreme points of $C([0,1])$ are the continuous functions that take values on the unit circle. However, I'm not sure how to reason from here as to whether or not it is ...
3
votes
1answer
31 views

Intuition behind the Riesz-Thorin Interpolation Theorem

Quoting the definition on Wikipedia, Let $(\Omega_1, \Sigma_1, \mu_1)$ and $(\Omega_2, \Sigma_2, \mu_2)$ be $\sigma$-finite measure spaces. Suppose $1 \leq p_0 \leq p_1 \leq \infty$, $1 \leq q_0 ...
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0answers
30 views

Proof of Hunt's Interpolation

I'm new to weak $L^p$ spaces and I'm doing a book exercise. Can someone enlighten me on the proof of the Hunt's interpolation theorem, which goes as follows: Theorem Let $\langle \,M, \mu \, ...
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0answers
24 views

a question about prove an exponential matrix function can be infinitely differentiable

If I have an exponential matrix exp(t(U+sH)), can someone tell me what is the dirivative with respect to s? I am really confused. (where U and H are matrices,and s,t are real numbers). Thus,if I let ...
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2answers
31 views

Extreme points of unit ball of Banach spaces $\ell_1$, $c_0$, $\ell_\infty$

Find extreme points of the unit balls of each Banach space, $l^1 $, $c_0$, $ l^\infty$ Can you help me with this one? For the first space, $l^1$, I thought there was no extreme point, but ...
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1answer
40 views

Why using liminf instead of limsup?

In Chapter 8: Calculus of variations of Evan's Partial Differential Equations, Evan writes as follows: I am wondering about the last paragraph where he says that knowing $I[u] \leq ...
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1answer
15 views

Numerical range of inverse operator

Let $T$ be a bounded self-adjoint operator such that the numerical range is contained in $[a,b]$ with $0<a<b< \infty.$ Does it then follow that the numerical range of $T^{-1}$ is contained in ...
2
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1answer
22 views

Continuity of functionals on Sobolev space

Let $U$ be a bounded set in $R^n$ and $W^{1,p}(U)$ denote a Sobolev space. Suppose $\{w_n\}\subset W^{1,p}(U)$ converges to $w \in W^{1,p}(U)$. Let $I[w]=\int_U F(Dw,w,x)dx$ for $w\in W^{1,p}(U)$, ...
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1answer
27 views

Closure of bounded set is bounded? Topological space

Let $X$ be topological space, and $A \subseteq X$ that is bounded. Is the closure of $A$ also bounded? This is true if $X$ is topological vector space, but is it if $X$ is only topological?
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0answers
33 views

Measureable functions and its properties [on hold]

I have two question about measureable functions and its properties and I want some help to solve them $1)$ if $f$ and $g$ are positive measureable functions then $f-g$ is measureable function ? ...
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0answers
20 views

Using Bounded Operator sequence Theorem

Let E$\subset L_1$ be a set of fourier series functions $e_n(t)=e^{int}$ for $n \in Z$. What is meant by saying to prove $Ge_n$ is a scalar multiple of $e_n$ and it is continuous? How can we prove it? ...
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1answer
42 views

Why every algebra on finite set is a topology

How can I prove that every algebra on finite set is a topology on this set And if the set is infinite how can give me an example algebra but it isn't topology
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1answer
16 views

How can I prove that in monotone class

How can I prove that : Let $X$ be nonempty set and $A$ is algebra in $X$ and $A$ is a monotone class , then $A$ is $σ$ Algebra in $X$
2
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0answers
33 views

How can I solve like this exercise about measurable function

How can I solve the following exercise Every positive measurable function is limit of increasing sequence of positive simple function How can I prove that I need the proof with explain or how can ...
1
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1answer
29 views

can a LUB be part of an open interval

I am re-reading an old textbook "Introduction to Hilbert spaces and applications" by Lokenath Debnath and Piotr Mikusinski, and there is a proof of a lemma in a chapter about the Lebesgue integral ...
0
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0answers
24 views

Weak convergence of a sequence of elements of a compact set implies strong convergence

Let E be a Banach space and $K \subset E$ a compact subset in the strong topology. Let $(x_n)_{n \geq 1} \subset K$ such that $x_n \rightharpoonup x$ in $\sigma (E, E^*)$. K is compact, so $(x_n)_n$ ...