Functional analysis

Which of these normed spaces are complete (Banach spaces)? Select all that apply.

The diagonal operator $T$ on $\ell^2$ is defined by $(Tx)_n = \dfrac{3n}{n+1}\, x_n$ for $n \ge 1$. Find $\|T\|$.

On $c_{00}$ (sequences with finitely many non-zero terms) with the sup norm, let $T_n(x) = n x_n$. Each $x$ has $\sup_n |T_n x| < \infty$, yet $\|T_n\| = n$. Why does this not contradict the uniform boundedness principle?

On $L^2(0, 1)$ let $\ell(f) = \int_0^{1/2} f(x)\, dx$. By the Riesz representation theorem $\ell(f) = \langle f, g \rangle$ for a unique $g$. What is $\|\ell\|$?

Which operator on $\ell^2$ is compact?

In the proof that a compact self-adjoint $T \ne 0$ has an eigenvalue $\lambda = \pm\|T\|$, we have unit vectors with $\|Tx_n - \lambda x_n\| \to 0$. Where is compactness used?

Which Lean statement correctly formalises "if a real inner product space $E$ contains an orthonormal sequence, its closed unit ball is not compact"?

Apply Gram–Schmidt to $1, x, x^2, \dots$ in $L^2(-1, 1)$, keeping the polynomials monic (leading coefficient $1$) instead of normalising. A polynomial is a list of Fraction coefficients, constant term first. The helper inner(p, q) computes $\int_{-1}^1 p q \, dx$ exactly. Write monic_orthogonal(n) returning the first n monic orthogonal polynomials (the monic Legendre polynomials).