Hi, thanks for the clarification - but there are still several questions:
It seems that your primary concern is
a) using a short piece of wire (<10 m?)
b) feed it from one end
c) for multiple bands, namely 40, 20, and 10m.
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I am not entirely sure what you mean by ‘resonant’. A monopole radiator or a dipole made from wire is initaelf typically resonant on a series of fractions of the wavelength. But resonance in this sense is not super-important, because you can design a matching circuit that will make sure that the output from your transmitter will see a proper 50 R impedance with no inductive nor capacitative component.
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What matters is the efficiency of the antenna, which will depend on several factors, namely the radiation resistance, the ground losses, and, often ignored, the fit of the pattern of radiation at the typical location of usage to the other stations you want to communicate with.
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Basically, you can use any piece of conductive material of any geometry as an antenna, even a wet shoestring or the leg of your barbecue table. Even a 50 R resistor is a kind of all-band antenna, albeit a very bad one.
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Now, you can use a combination of inductors, capacitors, traps, and mechanical tricks (e.g. folding back a part of the wire) to influence the amplitude and phase of current along your antenna at a given frequency. The best choice will depend on electrical and mechanical aspects (e.g. base loading coil vs. top-hat capacitor).
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The frequency-dependent behavior of inductors and traps (typicall LC parallel circuits) can be used to vary that behavior for different bands.
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If the antenna is a monopole, you need some other conductive structure so that a current can flow from the transmitter to your antenna. If it is a dipole, you may not need another pole. This depends on where you feed the antenna. For a center-fed antenna, you can split the element in half and use each half as one of the two poles. The split can also happen at other places, but there are dependencies with #4 and #5.
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The position of the feedpoint influences the raw feedpoint impedance. If you move the feedpoint along the antenna structure, the impedance will change. This is not a big deal, because you can transform any feedpoint impedance to a pure, resistive impedance of 50R for your transmitter. There may be losses, but not show-stoppers. Some matching techniques will be band-specific (e.g. an L-R match or hairpin match), others can be build in multi-band fashion, like an impedance autotransformer on a toroid.
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The extreme in the range of feedpoint position is an end-fed half wave antenna, where either end will work as a feedpoint witn a very high impedance; high enough, that very imperfect conductors in the system, connected to the ground pole of your transmitter, will be sufficien for the antenna to radiate.
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Now, you can e.g. build a multi-band EFHW with traps (popular for 40-30-20m), and the inductance of the LC circuit that makes the traps will work like a moderate loading coil and reduce the length of the wire sections that will produce resonance. To a certain degree, you can vary the inductance of each trap and compensate by adjusting the capacitor so that the resonant frequency of the trap remains unchanged.
However, there are limits to how far you can do so, because a trap is optimal only at a certain ratio of L and C. The losses from the trap will grow if you move away from that point for the sale of shorter wire sections.
Also, much more important, by shortening sections of wires, you change
a) the effective length of wire that radiates (the coil will almost not contribute),
and, more importantly,
b) you change the distribution of amplitude and curreny along the combined sections of wire, and this in different ways for different bands.
Now, you can insert additional inductors into the wire segments at your discretion in order to influence that, but the shorter the effective length of the antenna, the more difficult will this become, and the less efficient will the antenna be.
Roughly speaking, antennas down to ca. 50% of the ‘natural’ resonant length can be made with acceptable losses in efficienty, but beyond that point, it will become difficult.
Now, I know some people tried to construct shortened 3-band end-fed half-wave antennas by putting two inductors at carefully chosen lengths, which will then serve as both a means to compensate for the lacking length of wire and as traps for passing only lower frequencies onto the next segment.
I have one of these at home, but have not yet tested it. But I expect that this will
only work well at total lengths > 50% of the resonant EFHW/dipole length for the lowest band.
If you want to go shorter, my gut feeling is that going to a monopole plus some kind of radials, counterpoise, second leg (up-and-outer) will be more effective. The only downside of that path is that the antenna will typically not be resonant across differeny terrain, so you will need a tuner. But a drastically shortened dipole will also be much more prone to detune depending on deployment and terrain (if hanging low).
A long summary of what I learned about antennas in the last seven years 
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Experts - feel free to correct if needed.
73 de Martin, DK3IT
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