# Tagged Questions

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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### Proving that a subspace of a normed vector space is closed

Question: Let X be a normed vector space. If M is a closed subspace of X and x ∈ X − M then M + ℂx is closed where M + ℂx = { y + λ x : y ∈ M , λ ∈ ℂ } There's a theorem from Folland's Real Analysis ...
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### How to determine a function from its corresponding distribution?

If we have a function $\phi(x)$ we can determine the corresponding distribution $\phi^D$ such that: $$\forall f:L_{\phi^D}(f)=\langle\phi^D|f\rangle=\int_\mathbb{R}\phi(x) f(x) dx$$ as long as $f$ ...
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### Show that $g\in L^1(\mathbb{R})$ but $g^2\not\in L^1(\mathbb{R})$.

$$g(x) = \begin{cases} 1/\sqrt{x} & \text{ for }0<|x|<1,\\0 & \text{ otherwise.} \end{cases}$$ Show that $g\in L^1(\mathbb{R})$ but $g^2\not\in L^1(\mathbb{R})$. Wouldn't $g^2$ just be ...
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### A problem on real analysis in which from derivative you need to tell about function.

Let $f:(0,\infty)\to\mathbb{R}$ be differentiable.if $f'(x)\to l$ as $x\to\infty,$then show that $\frac{f(x)}{x}\to l$ as $x\to\infty$ . i have no idea where to start.any hint please
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### Are exponential functions a complete system on $L^2\left(\mathbb{T},\mu\right)$

Let $\mathbb{T}$ denote the unit circle in $\mathbb{C}$, and let $\mu$ be a complex Borel measure on $\mathbb{T}$. Consider the space of square integrable functions $L^2\left(\mathbb{T},\mu\right)$. ...
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### Bounded sequence in a normed space converges weakly

Can anyone help me here? Question: "X is a normed space and A is a subset dense in the dual of X. x belongs to X and the sequence (x_n) of X is bounded of E such that f(x_n) converges to f(x) for all ...
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### Dual Bases: Finite VS Infinite Dimensional Spaces

Motivation I know that in finite dimensional spaces like a $n$ dimensional Euclidean space $\mathbb{E}^n$, for every basis $G=\{g_1,g_2,...,g_n\}$ we can define a dual basis $G'=\{g^1,g^2,...,g^n\}$ ...
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### Show that if $f$ is a uniformly continuous function on $\mathbb{R}$ and $f\in L^1(\mathbb{R})$, then $f$ is bounded and $\lim_{|x|\to\infty}f(x)=0$.

Show that if $f$ is a uniformly continuous function on $\mathbb{R}$ and $f\in L^1(\mathbb{R})$, then $f$ is bounded and $\lim_{|x|\to\infty}f(x)=0$. I'm not entirely sure what I should be doing. ...
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### Prove that $l^2$ is closed and bounded but not compact.

Consider the space $l^p=\{(x_i);x_i\in \mathbb C:\sum |x_i|^2<\infty\}$ .Define a norm on $l^2$ by $||x||=\sqrt{\sum |x_i|^2}$. Prove that $l^2$ is closed and bounded but not compact. I know ...
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### If $A$ is a positive operator and $B$ is a bounded operator, show that $B^*AB$ is positive.

If $A$ is a positive operator and $B$ is a bounded operator, show that $B^*AB$ is positive. Where both $A$ and $B$ are operators in a Hilbert space $H$. I know that if $A$ is a positive operator, ...
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### Characterize the continuity of linear maps between Banach spaces in terms of continuous linear functionals

I solved a) and b). But I cant seem to get a grip of what characterizes the functionals for which we want continuity in the general case. Hints please!
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### Normed Quotient Space

If $X$ is a normed vector space and $M$ is a proper closed subspace, I want to show that for any $\epsilon>0$ there exists an $x\in X$ such that $\|x\|=1$ and $\|x+M\|\geq 1-\epsilon$. Is there ...
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### Closure of a differential operator

Consider $A:\mathcal{D}(A)\subset L^{2}[0,1]\to L^{2}[0,1]$ given by $$A:=-\frac{d^{2}}{dx^{2}},\qquad\mathcal{D}(A):=C^{2}_{0}(0,1)$$ Now, I assume that the closure of $A$ is its extension defined ...