# Questions tagged [functional-analysis]

Functional analysis, 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 and other topics. For basic questions about functions use more suitable tags like (functions), (functional-equations) or (elementary-set-theory).

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### Relationship between the Fundamental Lemma of Calculus of Variations and Completeness in Statistical Inference

I've been studying the Fundamental Lemma of Calculus of Variations and the concept of completeness in statistical inference, and I've noticed that both concepts involve the idea that an integral (or ...
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### I have a question about finite dimension norm linear spaces

Let R is linear space over the field Q and Q is a finite dimension subspace of R the what about the completeness of Q ?? Since we know that any finite dimension norm space is complete but here Q is ...
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### Why does a multiplicative functional $m \in \Delta_A$ extend uniquely to $m^e \in \Delta_{A^e}$?

The setup is as follows. $A$ is a commutative Banach algebra, and $\Delta_A$ is the structure space of $A$, consisting of all non-zero continuous algebra homomorphisms $f:A\to \Bbb C$. The text says ...
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### Show that $P$ is normal if and only if it is an orthogonal projection.

Let $V$ be a finite dimensional complex inner product space, and $P$ be a projection. Show that $P$ is normal if and only if it is an orthogonal projection. My work: For the statement "$P$ is ...
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### If $T$ is a positive operator, and with $S^2=T$, then $S$ must be self-adjoint.

Let $S$ and $T$ be a linear operator on finite dimensional vector space $V$. If $T$ is a positive operator, and with $S^2=T$, then $S$ must be self-adjoint. Does this statement hold? I try to use the ...
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### Show that finite dimensional topological subspaces are topologically isomorphic to $\mathbb{C}^n$ using induction.

Let $X$ be a topological Vector space and $Y$ a finite dimensional subspace with $\text{dim}Y = n$, $n \in \mathbb{N}$. Now, one can show that every Isomorphism between $\mathbb{C}^n$ and $Y$ is a ...
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Let $X$ be Banach space. Prove that $(Z^{\perp})^{\perp\perp} = \overline{Lin Z}$ and $\overline{Lin Y} \subset (Y^{\perp\perp})^{\perp}$, where $Z^{\perp}$ is annihilator of $Z \subset X$, and $Y^{\... • 161 3 votes 2 answers 59 views ### Weak convergence of functionals$g_n^*(f) = n\int_0^1 x^nf(x)dx$[closed] Show that sequnce of functionals$g_n^*(f) = n\displaystyle{\int_0^1 x^nf(x)dx}, f \in C[0,1]$converges weakly and find its limit functional. Does it converge in the norm of space$C^*[0,1]$? I don'... • 161 1 vote 1 answer 35 views ### Weak, strong and uniform convergence of operator on$L^p(\mathbb{R}^d)$[duplicate] For every$a \in \mathbb{R}^d$, let$T_a(f)(x) = f(x-a), \forall f \in L^p(\mathbb{R}^d), \forall x \in \mathbb{R}^d$. Prove that$T_a$is a linear isometry of space$L^p$on itself. Find$\lim_{a\to ...
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I'd like to find the weak formulation of the problem $-u''+au=f$ on $(0,1)$ $u(0)=0$ $u'(1)=b$ $a>0$ and show that there exists a unique solution using Lax-Milgram. What I did: By multiplying ...