Let $T$ be a countably infinite subset of $\left[0,1\right]$, and let $\left\{ c_{t}\right\} _{t\in T}$ be a sequence of complex numbers so that:
$$\sum_{t\in T}\left|c_{t}\right|^{2}<\infty$$ Let: $$\varphi_{T}\left(x\right)\overset{\textrm{def}}{=}\sum_{t\in T}c_{t}\delta\left(x-t\right)$$ where $\delta$ is the Dirac Delta distribution, and consider the action of $\varphi_{T}\left(x\right)$ on functions $f\left(x\right)$ given by:$$f\left(x\right)\mapsto\left\langle \varphi_{T}\mid f\right\rangle \overset{\textrm{def}}{=}\int_{0}^{1}\varphi_{T}\left(x\right)f\left(x\right)dx$$
1) First of all: am I allowed to use standard inequalities (Cauchy-Schwarz, Hölder, etc.) with integrals of this form? For instance, is: $$\int_{0}^{1}\left|\varphi_{T}\left(x\right)f\left(x\right)\right|dx\leq\left(\sqrt{\int_{0}^{1}\left|\varphi_{T}\left(x\right)\right|^{2}dx}\right)\left(\sqrt{\int_{0}^{1}\left|f\left(y\right)\right|^{2}dy}\right)$$ legal? Moreover, can I then write: $$\int_{0}^{1}\left|\varphi_{T}\left(x\right)\right|^{2}dx=\int_{0}^{1}\sum_{t\in T}\sum_{\tau\in T}c_{t}\overline{c_{\tau}}\delta\left(x-t\right)\delta\left(x-\tau\right)dx=\sum_{t\in T}c_{t}\overline{c_{t}}<\infty$$?
2) If (1) is legal, can I then use Cauchy-Schwarz to obtain $$\left|\left\langle \varphi_{T}\mid f\right\rangle \right|\leq\left\Vert f\right\Vert _{2}\left\Vert \varphi_{T}\right\Vert _{2}$$ where $\left\Vert \varphi_{T}\right\Vert _{2}=\sqrt{\sum_{t\in T}\left|c_{t}\right|^{2}}$, and thereby conclude that $\varphi_{T}$ is a continuous linear functional on $L^{2}\left(\left[0,1\right]\right)$? I'm troubled by the fact that, for any f for which $\left\langle \varphi_{T}\mid f\right\rangle \neq0$, replacing $f$ with a function $g$ that equals $f$ everywhere on $\left[0,1\right]$ except at $T$, where $g$ vanishes, produces $\left\langle \varphi_{T}\mid g\right\rangle =0$, even though $g$ and $f$ are identical as elements of $L^{2}\left(\left[0,1\right]\right)$, because they differ at a countable subset?
3) If (1) is legal, but the $L^{2}$ problem in (2) is insurmountable, can I still use (1) and argue:$$\sqrt{\int_{0}^{1}\left|f\left(x\right)\right|^{2}dx}\leq\sqrt{\sup_{x\in\left[0,1\right]}\left|f\left(x\right)\right|^{2}\left(\int_{0}^{1}dy\right)}=\sup_{x\in\left[0,1\right]}\left|f\left(x\right)\right|$$so that $\left|\left\langle \varphi_{T}\mid f\right\rangle \right|\leq\left\Vert \varphi_{T}\right\Vert _{2}\sup_{x\in\left[0,1\right]}\left|f\left(x\right)\right|$, which then shows that $\varphi_{T}$ is a continuous linear functional on the Banach space of (piece-wise?) continuous functions $f:\left[0,1\right]\rightarrow\mathbb{C}$ under supremum norm? Or what about on $L^{\infty}\left(\left[0,1\right]\right)$?
4) If (1) isn't legal, then what sort of inequalities can I use when working with singular objects like dirac deltas or $\varphi_{T}$? More generally, what would be an appropriate space of test functions upon which $\varphi_{T}$ would act as a continuous linear functional?
It turns out that what I've been looking for is $L_{\textrm{count}}^{2}\left(\left[0,1\right],\mathbb{C}\right)$, the space of all functions $f:\left[0,1\right]\rightarrow\mathbb{C}$ which are square-integrable with respect to the counting measure.
I can't use $\ell^{2}$, because that would only allow me to keep track of the coefficient values $c_{t}$, and—for my purposes—what I need is to be able to keep track of both the $c_{t}$s and the points $t\in\left[0,1\right]$ at which the $c_{t}$s are non-vanishing.
Using the counting measure, I can dispense with needing to treat $\varphi_{T}$ as a distribution. Instead, it becomes a perfectly ordinary element of the Hilbert space $L_{\textrm{count}}^{2}\left(\left[0,1\right],\mathbb{C}\right)$, and acts on other elements of that space in the natural way—via the inner product. This also makes all of my worries about inequalities moot: the inequalities will hold because I'm working over an ordinary Hilbert space, no distributions necessary.