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)

2
votes
1answer
102 views

Find a basis of $(L^2((-\pi,\pi), \mathbb{R}))^2$

I need to find a basis of $(L^2((-\pi,\pi), \mathbb{R}))^2$. I believe a basis of $L^2((-\pi,\pi), \mathbb{R})$ can be produced by the eigenfunctions of $\triangle$ (see L.C. Evans: Partial ...
1
vote
1answer
140 views

Spectral radius of an operator .

I would like to know the spectral radius of the operator $T_k$ from $C[0,1] \to C[0,1]$ : $$T_k x (t)= \int_0^1 k(t,s) x(s) ds$$ where $k(x,y)\colon [0,1]^2 \to \mathbb C$ is continuous. And ...
2
votes
3answers
203 views

Where does the iteration of the exponential map switch from one fixpoint to the 3-periodic fixpoint cycle?

In the answering of another question in MSE I've dealt with the iteration of $x=b^x$ where the base $b=i$. If I reversed that iteration $x=log(x)/log(i)$ then I run into a cycle of 3 periodic ...
4
votes
1answer
212 views

Spectrum of a right shift operator.

I have some doubts on the following problem : Let us consider $T : \ell^1(\mathbb N) \to \ell^1(\mathbb N) $by $(x_1,x_2..... ) \to (x_2, x_3 ........) $. I want to find the eigen values and ...
1
vote
1answer
95 views

Picard Condition (searching for an idea)

The so-called Picard-condition is: Let X,Y be Hilbertspaces and $T\colon X\to Y$ is a compact operator with singular value decomposition system $\left\{(\sigma_j,u_j,v_j)\right\}$. An element ...
3
votes
1answer
123 views

About a counterexample of an inequality?

I have known how to use the compactive argument to prove the inequality (1), i.e. $1\leqslant p<n$, $\Omega\subset R^n$ is a bounded domain,$\forall \varepsilon>0$, there is ...
1
vote
0answers
36 views

A question about a relationship of expressions got from change of variables/inner products

Suppose $F:L^2(S) \to L^2(T)$ is linear homeomorphism such that $F(v) = v \circ \mathcal{F}$ where $\mathcal{F}:T \to S$ is a diffeomorphism. Suppose $$\lVert F(v) \rVert_{L^2(T)} \leq C\lVert v ...
3
votes
1answer
363 views

Eigenvalues integral operator - general case

Let $T$ be an integral operator on $L^2([0,1])$, such that: $$ (Tf)(x) = \int_0^1K(x,y)f(y)dy, $$ with $K(x,y): [0,1]^2 \rightarrow \mathbb{R}$ continuous and $K(x,y) = K(y,x)$, $K(x,y)\geq0$ $ ...
2
votes
3answers
152 views

Inverse of inverse of function?

What is the inverse of inverse of a function (I assume the original function is invertible)? Is this the original function? Is it always true?
2
votes
1answer
210 views

Closure of the set of functions with respect to the $\infty$-norm

I want to show that the set of functions $X=\{f\in C[0,1]:f(0)=0\}$ is closed with respect to the $\infty$-norm. Suppose $(f_n)\subset X$ is a convergent sequence, now I need to find some function ...
1
vote
1answer
127 views

Compactness and boundedness of multiplication operator .

I need help with the following problem, If $1\le p < \infty$ , $(m_k)_{k\in \mathbb N}$and $K \subset \ell^p(\mathbb N)$ and $T: \ell^p(\mathbb N) \to \ell^p(\mathbb N)$ , define multiplication ...
4
votes
1answer
135 views

Generalizing the weak derivative

I am wondering about the weak derivative in time. We say f has a weak derivative f' if $$\int_0^T f\phi' = -\int_0^T f'\phi$$ for all $\phi \in C_0^\infty(0,T)$. This definition uses the $L^2$ inner ...
7
votes
1answer
296 views

Weak generator of Feller semigroup

Let $(T_t)_{t \geq 0}$ a Feller semigroup and define a linear operator $(A,\mathcal{D}(A))$ by $$\mathcal{D}(A) := \left\{u \in C_{\infty}(\mathbb{R}^d); \exists f \in C_{\infty} \forall x \in ...
2
votes
1answer
144 views

Convergence in norm independent of the choice of the norm.

When I read the proof of the Lie product formula in Reed Simon's book on functional analysis (which essentially reduces to showing $\left\Vert X_n - Y_n\right\Vert\rightarrow 0$ as $n\rightarrow 0$ ...
2
votes
2answers
120 views

‎‎If $A$ contains ‎an ‎idempotent $e‎$ (‎‎$‎e‎\neq ‎‎0,1‎‎$‎) , then $‎\Omega(A)‎$ ‎is ‎disconnected

If $A$‎ ‎be a‎ ‎unital ‎abelian ‎Banach ‎algebra ‎and ‎contains ‎an ‎idempotent $e$‎ ‎(that ‎is ‎‎$‎e=‎e‎^{‎2‎}‎‎$‎) ‎other ‎than $0$‎ ‎and $1$‎ ,‎ ‎then help me to show that ‎‎$‎\Omega(A)‎$ ‎is ...
2
votes
1answer
510 views

Isometric isomorphism of Hilbert spaces and orthonormal basis

If I have an isomorphism of two separable Hilbert spaces that preserves norms, does the isomorphism map orthnormal basis to orthonormal basis? I can't show it.
1
vote
1answer
58 views

Function for unique hash code

I am interested in finding $F(x,y)$, such that $x$ and $y$ $\in \mathbb Z^+$ and $F(x,y)$ is one to one function i.e., $F(x,y)$ is unique for any unique unordered pairs of $x$ and $y$. Regards, ...
0
votes
1answer
44 views

‎‎$‎\langle ‎(‎x_{n}‎)‎,(y_{n})\rangle=\sum_{‎1‎}^{‎\infty‎}\frac{‎‎x_{‎n‎}‎‎\bar{y_{‎n‎}}}{n^{2}}‎$‎‎ defines an inner product

Check ‎that ‎the ‎formula ‎‎$‎\langle ‎(‎x_{n}‎)‎,(y_{n})\rangle=\sum_{‎1‎}^{‎\infty‎}\frac{‎‎x_{‎n‎}‎‎\bar{y_{‎n‎}}}{n^{2}}‎$‎‎ defines an inner product ‎on ‎‎$‎\ell‎^{‎\infty‎}‎$‎,‎ ‎the space of ...
2
votes
1answer
297 views

Subset of $C[0,1]$ is nowhere dense

Let $E_n$ be $$E_n:=\{f\in C[0,1]\mid \text{exist } x_f\in[0,1] \text{ such that } |f(x)-f(x_f)|\leq n|x-x_f|,\, \forall x\in[0,1]\}.$$ How show that $E_n$ is nowhere dense, that is, $\mathrm{int}\ ...
2
votes
0answers
65 views

The proof of Open mapping principle (Lax)

I am reading the proof in Peter Lax book (page 169). By baire category we have some set $MB_i$ dense in some open set U. Then they translate those to origo by the form $M(B_n -x_0)$. And state that ...
1
vote
1answer
77 views

Parseval type identity

I have an orthonormal system of functions $$ U = \left\{ u_{\lambda}(x) \in L_{2}(\mathbb{R}_{+}) \mid \lambda \in \left\{-1,\ldots,-n\right\} \cup\mathbb{R}_{+} \right\} $$ such that for any $f,g ...
1
vote
0answers
100 views

Finding the spectral radius and spectrum .

I am solving the following question : If $k:[0,1]^2\to \mathbb C$ is continuous and $T_k : C[0,1] \to C[0,1]$ such that $$(T_kx)(t)=\int_0^t k(t,s)x(s) ds$$ Define $k_n: [0,1]^2\to \mathbb C$ ...
2
votes
1answer
122 views

Is it necessary to have a normed space for the Heine-Borel-Property to hold?

The Heine-Borel-Property says: A subset M is compact iff it is closed and bounded. It is well known that the euclidean space $\mathbb{R}^n$ has this property. In a narrower sense I found the ...
3
votes
0answers
88 views

Compute the continuous spectrum of an unbounded operator in $L^2(\mathbb{R}^2)^2$.

In "Béthuel, F. und J. C. Saut: Travelling waves for the Gross-Pitaevskii equation. I. Ann. Inst. H. Poincaré Phys. Théor., 70(2):147–238, 1999." the authors write on page 150, that one can easily ...
2
votes
0answers
113 views

Existence of non-zero $\sigma$ -finite $R^{(\alpha)}$-invariant Borel measure in $R^{\alpha}$

Let $R^{\alpha}$ be a vector space of all real-valued functions defined on a non-empty parameter set $\alpha$. Let $\cal{B}( R^{\alpha})$ denotes a Borel $\sigma$-algebra of subsets of $R^{\alpha}$ ...
3
votes
0answers
82 views

Seeing $1/x$ as a distribution

I have to show that by defining $$\langle u, f\rangle=\lim_{\varepsilon\rightarrow 0}\int_{-\infty}^{-\varepsilon}+\int_{\varepsilon}^{\infty}\frac{f(x)}{x}dx$$ with $f\in\mathcal{D}(\mathbb{R})$, ...
2
votes
1answer
318 views

A generalization of the Cauchy-Schwarz inequality to linear operators

If $A$ is an operator and $A \in \mathcal{B_{+}(X)}$ (the set of the positive operators) then the generalization of the Cauchy-Buniakowsky-Schwarz inequality holds: $$|\langle Ax,y\rangle| \leq ...
2
votes
3answers
89 views

Closure of a subspace of $l^\infty$

Let $X$ be the following subspace of $l^\infty$: $$ X=\mathrm{lin}\{e_n:n\in\mathbb{Z}^+\} $$ where $e_j$ has zeroes everywhere except for one in the $j$-th entry. I want to know what the closure of ...
2
votes
1answer
84 views

Question on equality with mollifier

So I have this homework. Note that $J_\epsilon$ is the standard mollifier. If $\Omega\in\mathbb{R}^n$ open and $u\in C(\Omega)$, show that $J_\epsilon * u\rightarrow u$ uniformly on every compact ...
2
votes
0answers
257 views

Determining the spectral representation of a operator

The spectral representation for a self-adjoint operator $T \in L(H)$ for H a Hilbert space is written as: $$ T = \sum_{\lambda \in \sigma(T)} \lambda \pi_{\lambda}, $$ where $\sigma(T)$ is the ...
2
votes
1answer
176 views

Is $M(X)$ (regular borel measures), the dual of $C_0(X)$ separable?

For $X$ locally compact (let's take $X=\mathbb{R}^d$), we know that the dual of $C_0(X)$ is $M(X)$, the space of regular borel measures on X. $C_0(X)$ is separable but is $M(X)$ separable? I have ...
1
vote
1answer
59 views

Transforming Orthonormal Basis to Higher Dimension

Assume $\psi_n(x),\;x\in\mathbb R$ is an $L_2(\mathbb R)$ complete orthonormal series. Let $f:\mathbb R^d\rightarrow\mathbb R$ be smooth enough. Is it true that if $$\sum_{i=1}^d \frac {\partial ...
2
votes
2answers
198 views

equivalence of norms and direct sum

Let $(X,\|\cdot\|_X) $ be an infinite dimensional Banach space. Suppose that you can write $X=V\oplus W$. Write $x=v+w$ and define $(V\oplus W,\|x\|_\oplus :=\|v\|_X+\|w\|_X)$. Show that ...
1
vote
2answers
73 views

Question on continuity of functions from $X\times X\rightarrow Y$

I am stuck on the following problem, which I do not believe to be so difficult. Let $X$ and $Y$ be Banach spaces. Let $f:X\times X\rightarrow Y$ be a function such that for any fixed $x_0$, ...
0
votes
0answers
132 views

Prove that if $L(S)$ is bounded, where $S$ is the unit sphere of $U$, then $L$ is Lipschitz.

Let $U$ and $V$ be normed linear spaces over $\mathbb{R}$, and $L : U \mapsto V$ a linear function. Prove that if $L(S)$ is bounded, where $S$ is the unit sphere of $U$, then $L$ is Lipschitz. There ...
3
votes
1answer
65 views

Spectrum of operator on canonical orthonormal system

Define the operator $T: l^2 \rightarrow l^2$ on the canonical orthonormal system $(e_k)_k$ by: $$ Te_k := \frac{e_k}{k} + \frac{e_{k+1}}{k+1}, $$ such that for $a\in l^2$: $$ T((a_i)_i) = (a_1, ...
7
votes
1answer
286 views

existence of a minimizer for functional

My problem is the following: Show that the mapping $u \rightarrow ||\nabla u||^2 + (fu,u)$ has a minimum $u$ in $M:=\{ w \in H^1(\Omega): ||w||=1\}$ . The function $f$ is in $L^\infty$. I dont see ...
1
vote
1answer
80 views

How to prove the density result?

How to prove that $C_c^\infty(R^n)$ is dense in $C^0(R^n)$,where the topology of $C^0(R^n)$ defined as follows $u_k\rightarrow 0$ in $C^0(R^n)$ if and only if $\forall K\subset\subset R^n$,$sup_{x\in ...
5
votes
1answer
258 views

checking if a function is positive using Fourier coefficients

Given a function $$f(x) = \sum_{k=0}^N a_k\ \sin(k\pi x)$$ defined over the region $S = [0, 1]$, is there some way to check if $f(x) \geq 0$ for all $x \in S$ using the coefficients $\{ a_k;\ k \leq N ...
1
vote
0answers
84 views

Is the trivial vector space considered a Banach space?

Is the normed vector space $\{0\}$ considered a Banach space? I am asking this question because Cartan's Differential Calculus implicitly assumes that the identity operator has norm 1 in Section 1.9. ...
3
votes
2answers
309 views

Function Spaces, why is the space of continuous functions (without necessarily differentiability) not important?

The space $C^0$ denotes the set of continuous and differentiable functions, the space $C^1$ the set of the continuous and differentiable functions which have a continuous and differentiable first ...
0
votes
1answer
95 views

i‎. ‎$B(X)=C(X)$ ii‎. ‎$C_{0}(X)=C(X)$ iii‎. ‎$C_{0}(X)=C_{c}(X)$

‎Explain‎, ‎when we have the following equalities‎ : ‎ ‎i‎. ‎$B(X)=C(X)$. ii‎. ‎$C_{0}(X)=C(X)$. iii‎. ‎$C_{0}(X)=C_{c}(X)$. Where: ‎$B(X)=\{f:X\to \mathbb C \mathrel| f \text{ is bounded}\}$, ...
3
votes
0answers
91 views

Integration of sine^2 w.r.t. some norm

Let $||x||$ be any norm over $\mathbb R^n$. Let $B_T$ the open ball with radius $T$ w.r.t. to our norm, i.e. all $x\in\mathbb R^n$ such that $||x||<T$. Let $n\in\mathbb N$. How much ...
1
vote
2answers
266 views

Trace class operators

I have a question concerning the definition of the square root of bounded linear operators. To introduce some notation: tr denotes the trace of linear operators and $\mathcal{L}(H)$ denotes the set of ...
2
votes
2answers
102 views

A sequence in $C([-1,1])$ and $C^1([-1,1])$ with star-weak convergence w.r.t. to one space, but not the other

The functionals $$ \phi_n(x) = \int_{\frac{1}{n} \le |t| \le 1} \frac{x(t)}{t} \mathrm{d} t $$ define a sequence of functionls in $C([-1,1])$ and $C^1([-1,1])$. a) Show that $(\phi_n)$ converges ...
3
votes
1answer
1k views

Orthogonal projection on the Hilbert space .

I want to prove the following: If $X$ is a Hilbert space and $Y$ is a closed subspace of $X$, then every $x\in X$ can be written as $x=y+z $ where $y\in Y$, $z \in Y^\perp$. The ...
2
votes
1answer
94 views

Existence and uniqueness of PDE with solutions in $W^{k,p}$ with $p \neq 2$?

I just realised that i have never seen the space $W^{k,p}$, $p\neq 2$, used in showing existence/uniqueness to some PDE. Usually books/lectures build up theory about $W^{k,p}$ (like certain compact ...
5
votes
1answer
143 views

Spectrum of a weakly compact operator

It is well known that the power of a weakly compact operator is compact. Is the spectrum of a weakly compact operator is the same as a compact operator?
-1
votes
1answer
96 views

$\| f \|=\vert f(0) \vert‎ + ‎\sup \vert f'(t) \vert$ defines a norm on $C^{1}[0,1]$

Does $$\| f \|=\vert f(0) \vert‎ + ‎\sup \vert f'(t) \vert$$ defines a norm on $C^{1}[0,1]$‎, ‎which is the space of (real) functions on $[0,1]$ with continuous derivative? explain in detail (Tanks ...
0
votes
1answer
68 views

‎If $\|f_{n}-f\|_{u}<\epsilon$‎, ‎then $\vert f(x) \vert<\epsilon$ if $x\notin \operatorname{supp}(f_{n})$? Why?

‎My question is of the proof of theorem: ‎If X is an LCH space‎, ‎$C_{0}(X)$=$\overline{C_{C}(X)}$ in uniform metricin ‎following.‎ ‎Proof‎: ‎Let $\{f_{n}\}\subseteq C_{C}(X)$ that ...