# Tagged Questions

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

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### What does it exactly mean that finite linear combination to be dense?

This phrase comes up over and over again when studying Hilbert space, and since I don't have the strongest background in linear algebra, the statement like "finite linear combination of elements in ...
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### Showing that a Hilbert Basis $(e_n)_{n \in \mathbb{N}}$ verifies $u= \sum (u,e_n)e_n$

The definition I have been given for a Hilbert Basis in a Hilbert Space $H$ over $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$ is: A sequence $(e_n)_{n \in \mathbb{N}}$ is an orthonormal basis if it ...
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### Let $H$ be a Hilbert space and $Φ≤H$ be equipped with a topology. Under which topology on $Φ^*$ is $H^*\ni f\mapsto\left.f\right|_Φ$ continuous?

Let $(H,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a Hilbert space and $\left\|\;\cdot\;\right\|$ be the norm induced by $\langle\;\cdot\;,\;\cdot\;\rangle$ $\Phi$ be a vector subspace of $H$ equipped ...
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### Bilinear forms defines inner product on Hilbert Space

I have difficulties understanding the reason why when I have a self adjoint linear operator $T : \mathcal{H} \rightarrow \mathcal{H}$, and know that $A\|f\|^2 \leq \langle Tf,f \rangle \leq B\|f\|^2$ ...
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### Proof of Anti-Linearity of Hermitian Conjugate

How can I prove that the adjoint operation/ Hermitian conjugate in anti-linear i.e $(\sum_{i} a_i A_i)^\dagger = \sum_{i} a_i^* A_i^\dagger$, where $A$ is any linear operator on a Hilbert space $V$. ...
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### Is the span of a compact, connected and infinite dimensional subset of $l^2$ open as a subset of the closed span?

Let $K$ be a compact, connected and infinite dimensional subset of $l^2$. Is it possible to prove that $\operatorname{span}(K) \setminus \{0\}$ is open in $\overline{\operatorname{span}(K)}$?
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### Is $\delta:C_2[0,1]\to \Bbb{R}, \delta(f)=f(0)$ discontinuous?

Is the functional $\delta:C_{2}[-1,1]\to \Bbb{R}, \delta(f)=f(0)$ with $L_2$ norm discontinuous? I understood I should show whether or not the functional is unbounded, but I can't seem to understand ...
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### Is there a condition along with Strong convergence which gives convergence in norm?

Let $\{T_n\}$ be a sequence of operators in a Hilbert Space and $T_n$ converges to an operator $T$ in the strong operator topology. Is there any assumption which I can put on $\{T_n\}$ so that the SOT ...
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### Spectrum of right shift operator in weighted $l2$ sequence space
Let $l_2(a)$ be a hilbert space defined with following inner product: $\langle x_n,y_n\rangle = \sum a^k x_k y_k$. (It's a weighted sequence space with the weights $\omega_i = a^i$). It's elements ...
Suppose $H$ is infinite dimensional Hilbert space. Let $A$ be a dense subset of $H$. How to prove that $\mathrm{dim}_{\mathrm{orth.}}\ H \le \mathrm{card}(A)$ ? When we have equality ? I need only ...