# Tagged Questions

For question involving Hilbert spaces, that is, complete normed spaces whose norm comes from an inner product.

3answers
1k views

### A Banach space that is not a Hilbert space

Can someone give me an example of a Banach space that is not a Hilbert space? I can't think of any because I don't know how to show one space that can not have inner product structure.
1answer
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### right inverse of surjective linear map on hilbert space exists iff kernel is complement subspace

Suppose that $L : H \to H'$ is a surjective continuous linear transformation between Hilbert spaces. Show that there exists a continuous linear transformation $S : H' \to H$ such that $LS = I$. ...
0answers
105 views

### Riesz basis $\{|t|^\alpha e^{int}\}_n$

The class of Riesz bases is very large. It is extremely difficult to exhibit at least one bounded basis for a Hilbert space that is not equvalent to an orthonormal basis. We mention without proof the ...
2answers
89 views

### Examples of spectral decompositions

I would like examples of spectral decompositions and how they are obtained for normal compact operators and normal non-compact operators on an infinite dimensional hilbert space. I have googled it, ...
1answer
56 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) ...
0answers
173 views

### The support of Gaussian measure in Hilbert Space $L^2(S^1)$ with covariance $(1-\Delta)^{-1}$

Let $\mu$ be Gaussian measure defined on Hilbert space $\mathcal{H}=L^2(S^1)$ ($S^1$ - circle) by formula $$\int e^{(f,g)} d\mu(f) = e^{-\tfrac{1}{2}(g,C g) }.$$ The covariance operator $C$ is ...
1answer
176 views

### The eigenvalues of a compact and self-adjoint operator on Hilbert space

Show that if $K$ is a compact self-adjoint operator on Hilbert space then it has either finitely many eigenvalues or a sequence of eigenvalues $\lambda_n\to 0$ as $n\to \infty$.
1answer
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0answers
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### Operators on a Hilbert space question

For a Borel measure $\mu$ define $\langle S_\mu x,y\rangle=\int_H\langle x,z\rangle \langle y,z\rangle \mu(z)$. An exercise in my book that I am reading says that I could find a $\mu$ s.t. $S_\mu$ ...
1answer
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### Dense subspace in a Hilbert space

Let $H$ be a Hilbert Space and $\{e_n\}_{n\in\mathbb{N}}$ an orthonormal basis. Now let $(x_n)$ be a sequence in $H$ satisfying $$\sum_{n=1}^{\infty}||x_n-e_n||^2<1.$$ Prove that ...
1answer
82 views

### Common orthogonal basis for $L^2$ and $H^1$

How can we obtain a common orthogonal basis for the space $L^2(U)$ and $H^1(U)$ for some bounded open subset of $\mathbb{R}^n$? That this can be done is mentioned in Evans's Partial Differential ...
1answer
61 views

### Complex Projective Line

How can I go about showing that a collection of all states is the complex projective line $CP^1$? All I understand at the moment is that an element in $CP^1$ is of the form ...
0answers
51 views

### Closed affine subspace of $\mathcal{L}^{2}$

Consider a Hilbert space $\mathcal{L}^{2}=\lbrace X: X-\text{real-valued random variable}, \mathbb{E}(X^{2})<\infty \rbrace$ with the inner product $\langle X,Y\rangle=\mathbb{E}(XY)$. Let ...
2answers
56 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 ...
2answers
44 views

### Adjoint of an operator question.

Let T be a normal operator. Prove that $\|T\|^{2n}=\|TT^*\|^n$ Has it got something to do with $\|T\|=\|T^*\|$?
2answers
59 views

### weak derivatives of exp(-|x|) and Hilbert Spaces

To which Hilbert Space (W^m,2) of R does the function exp(-|x|) belongs? I know its weak derivative is (-exp(-x) for x>0, exp(x) for x<0 and c0 (arbitrary) for x = 0). This weak derivative is in ...
1answer
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### 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 $L^2(I)$. ...
0answers
544 views

### Is a closure of subspace N and and orthogonal complement of this subspace N orthogonal?

Ok, there is something I do not understand about what I run into today in an online document. I know it might sound simple but I am so new to topology so I am having hard time to understand. As we ...
1answer
88 views

0answers
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### Continuity domain for momentum operator

The momentum operator in one dimension in quantum mechanics is $P=-i\frac{d}{dx}$ (with $\hbar=1$). Consider it as an operator on $L_2(0,2\pi)$, the space of square-integrable functions on $(0,2\pi)$. ...
2answers
274 views

### Derivative on Hilbert space

Please, on a Hilbert space what is the derivative of $\displaystyle\frac{x}{||x||}$ ? I know that it's equal to $\displaystyle \frac{1}{||x||}-\frac{\langle x,\cdot\rangle}{||x||^3} x$ but can I ...