0
votes
0answers
20 views

Does Hilbert space with countable dimensions exist? [duplicate]

If there is a Hilbert space with infinite dimensions, can it have countably infinite dimensions? And does Banach space with countable dimensions exist?
-1
votes
1answer
29 views

Need help proving the equivalence of two norms !

Hey I could use alot of help with this problem please! Let (X, <-,->) be a Hilbert space over R. Then, let A: X -> X be a linear operator. Suppose that A is symettric and positive definite. Let ...
0
votes
1answer
23 views

I want to show one norm is less than or equal to another norm on C([0,1])

Let $|| \ ||_1$ be the norm on $C([0,1])$ defined by $||f||_1 = \int_0^1|f(t)|dt$. a) Show that $||f||_1 \le ||f||_{[0,1]}$ b) Are $|| \ ||_1$ and $|| \ ||_{[0,1]}$ equivalent? For part a) I think ...
1
vote
0answers
25 views

Finding the norm of a linear operation.

I am reading A course in real analysis by John McDonald, on page 530, it says "it is easy to show $|||J|||=1$" where $J$ is the linear operation $J:C([0,1])\rightarrow C([0,1])$, defined by $J(f)(x) = ...
0
votes
2answers
32 views

showing $\inf \sigma (T) \leq \mu \leq \sup \sigma (T)$, where $\mu \in V(T)$

I am trying to prove the following: Let $H$ be a Hilbert space, and $T\in B(H)$ be a self-adjoint operator. Then for all $\mu \in V(T)$, $\inf \left\{\lambda: \lambda \in \sigma (T) \right\}\leq \mu ...
0
votes
1answer
21 views

An identity on direct sum of Hilbert spaces

Let $M_i$ are the set of smooth complex valued functions ($i=0,1,2,...$) $L^2(M_i)$ are Hilbert spaces on $M_i$ then can we say $$L^2(\bigoplus_{i=0}^\infty M_i)\cong \bigoplus_{i=0}^\infty ...
0
votes
1answer
15 views

equivalency of weak convergence and strong convergence for this family of sequences

Let $H$ be a Hilbert space and let $f_n \in H$ be a sequence of orthogonal elements i.e $<f_n,f_m>=0 $ if $n\ne m$. Define the element $F_N= f_1 + f_2 +...+ f_N$ for each $ N\in \mathbb N$. ...
0
votes
1answer
66 views

Hilbert space $L^{2}(0,\pi)$

I wanted to know how I should proceed if I wanted to prove that the closed subspace of $L^{2}(0,\pi)$ generated by {$\sin(kx): k=1,2,...$} coincides with $L^{2}(0,\pi)$. Thanks.
2
votes
1answer
30 views

Density result in Hilbert space

Assume that $b\in \mathbb{C}$ such that $0<\vert b \vert <1$. We consider the familly $f_{p}=\{1,b^{p},b^{2p},b^{3p},b^{4p},...,b^{np},...)$. How can one prove that $\operatorname{Span}(f_{p}, \ ...
1
vote
1answer
45 views

Is the space of continuous functions a Cauchy complete?

I am so new to functional analysis so I am looking for an answer of a confusion I am having right now in my mind because I have seen many different answers for the question I am gonna ask below. I ...
1
vote
0answers
21 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 ...
1
vote
0answers
47 views

The Hilbert space $\mathcal{H}_\eta$ and unitary correspondence with $L^2[a,b]$

The question I have is related to a problem in Stein and Shakarchi's Real Analysis, Chapter 4. The problem Let $\eta(t)$ be a fixed strictly positive continuous function $[a,b]$. Define ...
2
votes
3answers
118 views

help with showing completeness

Let $\left\{H_n\right\}_{n=1}^\infty$ be a sequence of Hilbert spaces and let $H=\left\{\left\{x_n\right\}:x_n\in H_n, \sum ||x_n||^2<\infty \right\}$. Define the inner product as ...
0
votes
0answers
27 views

Range of $S$ is orthogonal to the kernel of $L$

Suppose that $L: H \to H' $ is a subjective continuous linear transformation between Hilbert spaces. If $S: H' \to H$ is also a continuous linear transformation such that $ LS = I$. Show that the ...
0
votes
1answer
25 views

Question about surjective continous operator being right invertible

I am reading a proof that a surjective continuous linear operator $T$ on a Hilbert space $H$ is right invertible. I have a question about the proof. The proof (up to the point where I have a question) ...
2
votes
2answers
44 views

Show that $\ell^2(A)$ and $\ell^2(B)$ are isomorphic iff $A$ and $B$ have the same cardinality

Let $A,B$ be sets. Show that $\ell^2(A)$ and $\ell^2(B)$ are isomorphic iff $A$ and $B$ have the same cardinality. (Here $\ell^2(A)$ is the square integrable functions that stand on $A$ with the ...
0
votes
1answer
40 views

Show that H$(I)$ is a closed subspace of $L^2(I)$

EDIT: Original statement is not true, added condition. Let $I$ be the unit interval, define $H(I) = \{f\in AC(I)$ and $f'\in L^2(I)\}$. I want to show that $H(I)$ a closed subspace of $L^2(I)$. ...
0
votes
1answer
34 views

Counterexample of minimum principle in hilbert space on non closed but convex subspace

As I mentioned at title, I make tiny counterexample for minimum principle. Let $K=C([0,\frac{1}{2}]) \subset H=L^{2}([0,1])$. Then $K$ is convex since every $f,g \in K$, $(\alpha ...
0
votes
0answers
32 views

Unconditionally convergence in Hilbert space

Let $H$ be a complex Hilbert space and $e_1, e_2,...$ be a countable orthonormal system in $H$, $c_1, c_2,...$ is a sequence of complex numbers. How to prove that if $c_n$ is square-summable then ...
2
votes
0answers
60 views

What's this standard duality argument?

I'm reading a proof of the Strichartz inequalities. It shows that $$ \| \int_\mathbb{R} e^{-is\Delta}F(s) \, ds \|_{L^2_x} \lesssim \|F\|_{L^{q'}_t L^{r'}_x}, $$ and then says that by duality, $$ ...
6
votes
1answer
45 views

Derive Fourier transform from what it should do?

I was wondering about the following: Imagine you want to figure out whether there is a transform that exchanges differentiation with multiplication and convoution with pairwise transformation for ...
0
votes
1answer
76 views

Proof Riesz representation theorem

I have a question regarding the proof of the Riesz representation theorem. Why do we declare the isomorphism $\Phi: H \rightarrow H'$ in an antilinear way? I mean if, this isomorphism would pick the ...
2
votes
1answer
34 views

Proof of an equivalence in Hilbert spaces

Let $H$ be a Hilbert space. Prove that the following are equivalent: a) the algebraic dimension of $H$ is finite; b) each closed, not empty subset $C$ has an element of minimum norm (that is the ...
4
votes
2answers
103 views

Gram-Schmidt in Hilbert space?

EDIT: After some contemplation I decided to phrase the question better to avoid trivial answers. Consider a Hilbert space with a basis $\{v_{i}\}$ where $i\in I$ an index set, which could be ...
0
votes
0answers
45 views

Verify that the operator $T$ defined by $T( \varphi _{k})=\frac{1}{k}\varphi _{k+1}$ is compact, but has no eigenvectors.

Let $H$ be a Hilbert space with basis $\left \{ \varphi _{k} \right \}_{k=1}^{\infty }$ .Verify that the operator $T$ defined by $$T( \varphi _{k})=\frac{1}{k}\varphi _{k+1}$$ is compact, but has no ...
0
votes
1answer
45 views

showing that a sequence is converging.

suppose $\left \{ T_{k} \right \}$ is a collection of bounded operators on Hilbert space $H$ ,with $\left \| T_{k} \right \|\leq 1$ for all $k$ .suppose also that $$T_{k}T_{j}^{*}=T_{k}^{*}T_{j}=0 ...
6
votes
1answer
94 views

Dense subspace of $L^{2}[0,1]$

I know that $C[0,1]$ is dense in $L^{2}[0,1]$ but is $\{f\in C^{2}[0,1]:f(0)=f(1)=0\}$ dense in $L^{2}[0,1]$?
0
votes
1answer
79 views

what is the advantage of having countable dense subset?

what is the advantage of having countable dense subset (for example of the set $L^2([0,1])$, if i have to prove weak convergence ? edit: to prove is that every sequence $(f_n)_n$ with ...
1
vote
1answer
71 views

Duals of Hilbert Subspace

So I am confused about something very basic. I'm going to outline my confusion, and would love if someone could point out when I'm saying something wrong. Let $H$ be a Hilbert space. It's dual $H^*$ ...
3
votes
2answers
93 views

A dense subset of a Hilbert space

I am curious about the following problem: Consider the Hilbert space (a weighted $L^2(\mathbb{R})$ space): $$\mathscr{H}=\bigg\{f: \mathbb{R}\to\mathbb{R}\text{ Lebesgue ...
3
votes
0answers
57 views

An orthonormal basis for a Hilbert space

Can anyone give me some hint on the following problem without using any knowledge about complex analysis or Fourier analysis? Thanks a lot! Consider the Hilbert space $$\mathscr{H}:=\bigg\{f\text{ ...
0
votes
0answers
19 views

inequality related to transformations and inner products

Let $T$ be a bounded transformation from a hilbert space to itself. Suppose that if $||f||\leq 1$ and $||g||<1$ then $|\text{Re}(Tf,g)|\leq M$ where we are taking the real part of the inner ...
1
vote
3answers
205 views

From analysis of realvalued functions to analysis of Hilbert/Banach-valued functions

Does anybody know of a text (doesn't matter which form: article, book etc. - anything's welcome) in which it is described which result from real analysis also hold for Hilbert/Banach spaces ? I'm ...
2
votes
1answer
80 views

Weighted $\ell^2$ space is Hilbert

This is my exam's question that I could not solve it. Please help me to undrestand how to solve it: let $\{w_n\}$ is a positive real numbers sequence, and let $$\ell^2(w):=\left\{\{x_n\}:x_n \in R ...
5
votes
0answers
118 views

Is there an orthonormal basis for $L_2[0,1]$ consisting of convex functions?

Is there an orthonormal basis $\{\phi_{\alpha}\}$ for the space $L_2[0,1]$ of square-integrable functions from $[0,1]$ to $\mathbb{R}$ such that every $\phi_{\alpha}$ is convex? Edit: A helpful ...
0
votes
0answers
49 views

Continuous and dense embeddings and the density of sets in Hilbert space.

Suppose $H$ is a Hilbert space of functions $f:\Omega\to \mathbb{R}^n$ with $\Omega\subset \mathbb{R}^n$ open, bounded and with Lipschitz boundary (take for example $H=H_0^1(\Omega)^n$) and suppose ...
5
votes
1answer
197 views

Prove or disprove this argument

Let $L>0$ and let $\Omega$ be the set of all integrable functions from $[0,L]$ to $]0,+\infty[$. For all $\varphi, \psi \in \Omega$ define $\left \langle \varphi,\psi \right ...
5
votes
1answer
135 views

A baby version of the Stein-Cotlar almost-orthogonality lemma

The following is an exercise from Stein and Shakarchi's Real Analysis. Suppose $\{T_k\}$ is a collection of bounded operators on a Hilbert space $H$, each with norm at most $1$. Suppose also that ...
1
vote
0answers
64 views

Are the special functions independent?

maybe the bessel functions are some complicated function of the exponential function, logarithm function... or maybe there's a relation between two or more transcendental functions. Is there a way to ...
0
votes
0answers
258 views

Haar functions are an orthonormal basis of $L^2[0,1]$ [duplicate]

The Haar functions are defined by $e_0^0(x)=1$, and for $n\geq1$ y $1\leq k\leq2^n$, $$e_n^k(x)=\left\{\begin{array}{rcl} 2^{\frac{n}{2}} & \hspace{0.125cm} & \text{if }\frac{k-1}{2^n}\leq ...
2
votes
2answers
155 views

Proof of the Riesz Representation Theorem

Theorem: Let $F$ be a continuous linear functional on the Hilbert space $H$, then $\exists !$ (exists one and only one) $y \in H$ such that $F(x) = (x,y)$ for $x\in H$. Proof: Uniqueness: ...
1
vote
1answer
81 views

Orthogonal family in Hilbert Space

Let $(x_k)_1^\infty$ be an orthogonal family of points in X a Hilbert space. Then $\sum_{i=1}^\infty x_i$ converges if and only if $\sum_{i=1}^\infty ||x_k||^2$ converges. Also need to show that ...
1
vote
0answers
55 views

Verify solution: Is this gradient, correct?

For a function $$f(X)=\operatorname{tr}(X^TAX)+\|\operatorname{diag}(X^TX)-\alpha I\|_2,$$ where all entries are real and $\alpha$ is a real scalar, while $A$ is a p.s.d matrix and $X$ is a real ...
2
votes
0answers
44 views

Prove that the sequence is in $\ell^{2}$. [duplicate]

Let $(a_{n})$ be a sequence of complex numbers such that for every $(b_{n})\in \ell^{2}$the series $\sum_{1}^{\infty}a_{n}b_{n}$ converges. Prove that $(a_{n})\in \ell^{2}.$ What I've tried so far is ...
2
votes
0answers
52 views

Find a bounded function with a supporting point

Given, $g(Z)=\operatorname{tr}\phi(Z)$, where $\phi(Z)= Z^T\left( \operatorname{diag}(ZZ^T\mathbf{1}) - ZZ^T\right) Z$ where $Z$ is a real rectangular matrix with more rows than columns (tall and ...
8
votes
1answer
221 views

Is my statistician friend right/wrong on metric spaces and norms?

I was talking to a statistician friend of mine who said that instead of minimizing this function $\sum_{i,j}W_{ij}d_{ij}^2(X)$ over $X$ it would be better to solve an analogous minimization problem ...
0
votes
1answer
196 views

Proving that Uniform operator convergence implies strong operator convergence implies weak

Let $H$ be a Hilbert Space. Let $\{T_n\}$ be a sequence of bounded operators in $H$. I'm trying to prove that Uniform Operator Convergence implies Strong Operator Convergence implies Weak Operator ...
0
votes
1answer
373 views

Density and closedness of $C[0,1]$ in $L^\infty[0,1]$ in norm and weak-* topologies

With results: "For convex subsets of a locally convex space, a, originally( strongly) closed equals weakly closed, and b, originally (strongly dense equals weakly dense." Could you help me solve this ...
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 ...
4
votes
2answers
398 views

Norm inequality for sum and difference of positive-definite matrices

If $X_{1}$ and $X_{2}$ are positive definite matrices, how to show that $\left\Vert X_{1}-X_{2}\right\Vert \le\left\Vert X_{1}+X_{2}\right\Vert$ for the spectral norm? and how about for the nuclear ...